messages listlengths 1 1 | ground_truth listlengths 15 15 | dataset stringclasses 1 value | id stringlengths 5 8 | difficulty int64 2 2 | timestamps listlengths 15 15 | scene_config dict | tolerance float64 50 50 | prompt_setting stringclasses 1 value |
|---|---|---|---|---|---|---|---|---|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (480, 738); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (495, 804); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (704, 725), initial velocity (-408.47, -622.18) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[672.3, 658.38]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[693.55, 613.3]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[766.59, 616.25]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[839.44, 631.53]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[826.99, 696.06]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[803.83, 766.79]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[780.66, 837.52]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[757.49, 877.52]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[734.32, 806.8]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[711.15, 736.07]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[687.98, 665.34]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[664.81, 619.62]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[707.09, 658.68]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[776.85, 684.65]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[846.6, 710.63]]) <= 50.0"
] | ballsim | F_2_47 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
704,
725
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-408.47,
-622.18
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
480,
738
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
495,
804
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 47,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 2744816
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (636, 985); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (617, 529); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (723, 721), initial velocity (555.22, -496.77) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[792.32, 731.78]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[863.9, 752.46]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[883.97, 748.3]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[823.18, 705.23]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[792.9, 676.86]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[864.48, 697.54]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[877.0, 746.65]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[848.53, 815.5]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[820.06, 884.35]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[811.57, 816.51]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[807.35, 742.12]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[803.12, 667.73]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[819.94, 622.35]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[856.02, 687.53]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[892.1, 752.72]]) <= 50.0"
] | ballsim | F_2_1018 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
723,
721
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
555.22,
-496.77
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
636,
985
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
617,
529
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1018,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6790306
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (1026, 703); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (611, 520); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (787, 750), initial velocity (-483.4, 537.05) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[738.66, 803.71]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[690.32, 857.42]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[703.06, 875.93]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[773.74, 860.91]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[844.42, 845.9]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[841.52, 788.39]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[802.97, 729.03]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[754.63, 675.32]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[801.77, 678.46]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[790.37, 658.2]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[785.92, 641.95]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[825.89, 651.33]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[801.64, 623.68]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[810.18, 614.93]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[769.57, 660.5]]) <= 50.0"
] | ballsim | F_2_288 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
787,
750
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-483.4,
537.05
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1026,
703
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
611,
520
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 288,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6784579
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (825, 503); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (635, 978); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (748, 700), initial velocity (-243.87, -646.16) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[677.55, 693.14]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[630.35, 696.0]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[672.81, 750.7]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[715.17, 761.01]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[752.24, 713.07]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[708.47, 766.49]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[664.69, 746.36]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[620.92, 692.95]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[684.63, 701.56]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[752.8, 712.74]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[783.39, 771.03]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[809.47, 835.0]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[825.88, 892.51]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[829.11, 841.73]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[855.19, 777.76]]) <= 50.0"
] | ballsim | F_2_617 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
748,
700
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-243.87,
-646.16
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
825,
503
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
635,
978
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 617,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6467083
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (605, 963); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (998, 659); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (705, 770), initial velocity (215.16, 603.17) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[747.86, 736.77]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[791.14, 693.07]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[729.21, 710.06]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[667.45, 727.0]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[605.68, 743.94]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[664.73, 760.89]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[726.5, 777.83]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[788.27, 794.77]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[850.03, 811.72]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[850.51, 864.04]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[815.02, 841.49]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[773.54, 792.69]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[732.17, 744.0]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[690.69, 695.2]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[649.21, 646.4]]) <= 50.0"
] | ballsim | F_2_975 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
705,
770
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
215.16,
603.17
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
605,
963
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
998,
659
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 975,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 8612592
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (874, 522); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (877, 509); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (719, 738), initial velocity (644.55, 178.44) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[783.45, 755.85]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[847.91, 773.69]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[884.02, 777.86]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[829.89, 738.6]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[775.77, 776.88]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[721.64, 816.14]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[667.52, 855.39]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[678.56, 812.93]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[704.81, 751.4]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[731.05, 689.88]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[695.65, 705.66]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[641.52, 744.92]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[621.11, 784.08]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[675.23, 823.34]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[729.36, 862.59]]) <= 50.0"
] | ballsim | F_2_583 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
719,
738
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
644.55,
178.44
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
874,
522
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
877,
509
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 583,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5162958
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (904, 984); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (494, 801); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (788, 758), initial velocity (427.65, 494.14) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[803.78, 773.57]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[765.14, 720.88]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[726.39, 668.07]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[687.74, 615.38]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[649.1, 651.54]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[711.05, 655.79]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[776.36, 658.42]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[841.67, 661.06]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[844.25, 699.9]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[813.88, 757.76]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[783.51, 815.62]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[722.1, 834.56]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[708.51, 868.45]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[723.62, 852.49]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[721.35, 787.18]]) <= 50.0"
] | ballsim | F_2_803 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
788,
758
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
427.65,
494.14
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
904,
984
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
494,
801
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 803,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4748464
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (802, 1010); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (531, 612); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (743, 760), initial velocity (-541.11, 455.16) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[688.9, 805.51]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[647.6, 843.62]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[712.13, 816.82]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[750.41, 757.37]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[788.7, 697.92]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[826.98, 638.47]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[791.78, 621.45]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[762.27, 648.64]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[794.6, 711.53]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[826.92, 774.42]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[869.61, 791.94]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[908.88, 740.79]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[843.93, 713.32]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[778.81, 685.78]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[737.12, 671.76]]) <= 50.0"
] | ballsim | F_2_537 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
743,
760
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-541.11,
455.16
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
802,
1010
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
531,
612
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 537,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 9639973
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (661, 1007); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (478, 693); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (785, 762), initial velocity (503.26, 489.36) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[835.32, 810.93]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[847.4, 841.44]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[793.77, 812.29]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[769.65, 746.38]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[745.53, 680.46]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[721.41, 614.54]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[749.24, 613.76]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[808.16, 651.94]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[867.07, 690.13]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[891.18, 745.06]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[883.38, 814.47]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[833.06, 863.39]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[846.18, 847.63]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[809.68, 787.67]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[773.18, 727.71]]) <= 50.0"
] | ballsim | F_2_700 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
785,
762
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
503.26,
489.36
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
661,
1007
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
478,
693
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 700,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 8050590
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (624, 994); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (663, 992); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (766, 750), initial velocity (144.63, 781.57) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[789.28, 789.5]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[836.23, 725.37]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[850.45, 680.03]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[771.42, 688.63]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[692.39, 697.22]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[615.92, 707.29]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[649.05, 775.4]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[709.5, 723.79]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[769.95, 672.17]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[830.4, 620.56]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[766.89, 640.53]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[691.97, 667.09]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[624.57, 697.83]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[639.22, 775.14]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[686.16, 711.01]]) <= 50.0"
] | ballsim | F_2_957 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
766,
750
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
144.63,
781.57
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
624,
994
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
663,
992
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 957,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 803915
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (698, 998); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (695, 500); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (690, 787), initial velocity (249.73, -565.56) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[715.33, 728.86]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[752.61, 726.19]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[799.78, 766.4]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[846.82, 806.51]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[852.61, 826.76]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[817.37, 783.07]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[792.39, 726.5]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[778.06, 716.55]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[780.11, 778.34]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[796.86, 775.66]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[825.5, 720.88]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[854.14, 666.09]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[870.73, 663.94]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[872.78, 725.74]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[874.83, 787.53]]) <= 50.0"
] | ballsim | F_2_940 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
690,
787
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
249.73,
-565.56
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
698,
998
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
695,
500
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 940,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4387873
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (684, 1005); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (693, 494); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (737, 693), initial velocity (568.46, -217.13) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[793.84, 712.97]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[850.67, 734.67]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[907.65, 756.43]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[862.5, 719.25]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[815.29, 680.87]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[839.1, 731.58]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[866.0, 786.17]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[844.05, 812.55]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[783.34, 816.56]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[730.3, 786.87]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[677.32, 756.94]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[624.35, 727.0]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[644.36, 739.2]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[696.78, 770.11]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[749.19, 801.02]]) <= 50.0"
] | ballsim | F_2_800 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
737,
693
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
568.46,
-217.13
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
684,
1005
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
693,
494
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 800,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5640865
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (504, 652); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (858, 497); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (795, 775), initial velocity (555.84, -567.8) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[850.59, 718.21]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[891.57, 743.43]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[871.46, 818.31]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[810.69, 869.5]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[749.93, 865.07]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[689.16, 813.87]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[683.47, 794.48]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[758.17, 821.54]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[832.88, 848.59]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[809.89, 819.21]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[749.13, 768.01]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[693.2, 719.61]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[767.72, 746.6]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[842.43, 773.65]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[870.01, 773.48]]) <= 50.0"
] | ballsim | F_2_792 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
795,
775
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
555.84,
-567.8
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
504,
652
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
858,
497
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 792,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 8355672
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (690, 996); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (517, 890); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (700, 727), initial velocity (604.96, -481.99) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[760.5, 678.81]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[821.0, 630.61]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[861.27, 671.57]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[894.86, 741.24]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[858.96, 777.46]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[783.55, 794.65]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[717.38, 771.32]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[661.96, 720.19]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[650.46, 643.7]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[721.43, 670.6]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[793.45, 698.82]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[865.47, 727.04]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[904.95, 770.6]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[861.4, 834.54]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[839.35, 830.34]]) <= 50.0"
] | ballsim | F_2_325 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
700,
727
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
604.96,
-481.99
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
690,
996
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
517,
890
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 325,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5865829
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (1000, 776); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (715, 1002); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (721, 746), initial velocity (70.39, -774.13) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[728.03, 668.58]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[735.06, 623.07]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[742.11, 700.68]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[749.14, 778.09]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[756.17, 758.74]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[763.21, 681.32]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[770.24, 610.33]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[777.27, 687.74]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[784.3, 765.16]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[791.33, 771.69]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[790.24, 694.27]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[783.21, 616.86]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[776.2, 674.6]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[769.16, 752.02]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[762.13, 784.84]]) <= 50.0"
] | ballsim | F_2_580 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
721,
746
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
70.39,
-774.13
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1000,
776
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
715,
1002
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 580,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4797106
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (471, 752); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (643, 995); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (730, 787), initial velocity (222.43, 703.32) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[752.24, 742.93]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[774.48, 672.59]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[796.78, 612.15]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[819.02, 682.49]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[841.26, 752.82]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[863.5, 823.16]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[805.35, 847.06]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[821.0, 812.34]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[870.79, 757.92]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[871.93, 731.58]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[799.91, 747.47]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[727.89, 763.37]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[699.64, 800.05]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[732.11, 733.81]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[764.58, 667.58]]) <= 50.0"
] | ballsim | F_2_967 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
730,
787
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
222.43,
703.32
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
471,
752
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
643,
995
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 967,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 9807522
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (820, 1005); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (594, 962); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (764, 805), initial velocity (300.52, -662.08) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[794.05, 738.79]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[824.11, 672.58]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[846.94, 638.15]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[845.28, 710.83]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[843.62, 783.51]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[841.96, 764.08]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[840.3, 691.4]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[838.33, 620.09]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[805.3, 684.86]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[772.26, 749.63]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[739.23, 805.86]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[706.2, 741.09]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[673.25, 676.48]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[678.32, 659.48]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[734.13, 706.09]]) <= 50.0"
] | ballsim | F_2_576 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
764,
805
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
300.52,
-662.08
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
820,
1005
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
594,
962
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 576,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 3787591
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (881, 526); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (536, 592); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (780, 803), initial velocity (-278.32, -590.75) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[752.17, 743.92]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[724.34, 684.84]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[715.52, 697.79]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[697.49, 758.17]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[688.31, 822.82]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[686.18, 878.63]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[747.16, 855.29]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[808.15, 831.95]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[869.14, 808.61]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[877.97, 760.16]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[858.2, 743.8]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[838.42, 806.03]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[818.7, 868.11]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[781.25, 852.95]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[736.43, 805.46]]) <= 50.0"
] | ballsim | F_2_988 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
780,
803
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-278.32,
-590.75
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
881,
526
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
536,
592
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 988,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6871945
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (693, 1014); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (656, 998); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (796, 775), initial velocity (172.09, -593.64) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[813.21, 715.62]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[830.42, 656.25]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[837.66, 640.63]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[827.4, 701.59]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[817.15, 762.55]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[806.9, 823.52]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[863.95, 800.08]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[890.43, 761.75]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[873.22, 702.37]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[856.0, 643.0]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[823.97, 648.58]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[782.71, 694.6]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[741.56, 740.5]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[700.3, 786.52]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[659.04, 767.2]]) <= 50.0"
] | ballsim | F_2_841 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
796,
775
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
172.09,
-593.64
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
693,
1014
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
656,
998
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 841,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 9610363
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (506, 815); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (618, 992); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (778, 809), initial velocity (639.59, 180.91) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[841.96, 827.09]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[837.08, 805.44]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[789.31, 758.99]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[741.65, 712.65]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[694.0, 666.31]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[666.95, 631.87]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[730.92, 649.96]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[794.88, 668.06]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[858.85, 686.15]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[871.44, 733.89]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[855.13, 798.32]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[838.82, 862.75]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[822.55, 858.75]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[806.24, 794.32]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[789.93, 729.89]]) <= 50.0"
] | ballsim | F_2_370 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
778,
809
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
639.59,
180.91
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
506,
815
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
618,
992
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 370,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 1083443
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (873, 974); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (685, 502); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (791, 807), initial velocity (756.77, -175.37) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[810.24, 718.69]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[844.17, 648.82]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[859.44, 675.44]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[902.99, 739.77]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[852.35, 749.73]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[774.86, 744.19]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[697.37, 738.65]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[619.88, 733.1]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[625.76, 775.69]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[669.32, 840.02]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[649.19, 824.5]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[596.16, 767.74]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[639.57, 733.29]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[715.25, 715.77]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[790.93, 698.24]]) <= 50.0"
] | ballsim | F_2_28 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
791,
807
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
756.77,
-175.37
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
873,
974
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
685,
502
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 28,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 1241401
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (1006, 813); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (833, 999); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (755, 773), initial velocity (-704.02, 102.51) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[684.59, 783.25]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[614.92, 793.08]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[659.1, 737.1]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[703.17, 681.27]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[747.23, 625.43]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[791.3, 651.17]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[814.87, 681.29]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[770.24, 625.9]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[725.61, 650.27]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[680.98, 705.67]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[636.36, 761.06]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[646.02, 785.11]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[716.14, 774.17]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[786.43, 763.21]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[743.89, 752.25]]) <= 50.0"
] | ballsim | F_2_727 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
755,
773
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-704.02,
102.51
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1006,
813
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
833,
999
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 727,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 88555
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (1020, 775); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (605, 518); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (754, 759), initial velocity (652.11, -368.89) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[819.21, 722.11]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[824.64, 650.7]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[823.98, 645.18]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[823.32, 720.1]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[760.44, 759.1]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[695.89, 797.14]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[642.43, 828.78]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[707.64, 791.89]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[772.85, 755.0]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[819.35, 707.3]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[818.7, 632.38]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[818.04, 663.31]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[812.43, 735.19]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[747.88, 773.23]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[683.33, 811.27]]) <= 50.0"
] | ballsim | F_2_38 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
754,
759
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
652.11,
-368.89
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1020,
775
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
605,
518
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 38,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 1221787
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (507, 839); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (487, 687); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (742, 729), initial velocity (-396.65, -628.94) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[702.33, 666.11]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[701.7, 626.81]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[764.9, 666.36]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[827.93, 705.81]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[890.97, 745.27]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[863.52, 741.14]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[793.38, 716.46]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[723.24, 691.78]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[764.31, 720.65]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[827.35, 760.11]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[890.39, 799.56]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[821.44, 775.45]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[751.47, 750.83]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[700.53, 735.39]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[763.57, 774.85]]) <= 50.0"
] | ballsim | F_2_984 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
742,
729
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-396.65,
-628.94
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
507,
839
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
487,
687
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 984,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 1830724
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (982, 869); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (690, 504); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (729, 704), initial velocity (-575.44, 248.03) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[671.46, 728.8]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[613.91, 753.61]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[646.48, 778.48]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[704.02, 803.28]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[761.56, 828.09]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[733.5, 852.89]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[675.95, 877.69]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[684.03, 820.21]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[695.41, 758.59]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[706.78, 707.3]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[718.16, 768.92]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[729.54, 830.54]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[740.89, 892.01]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[729.85, 855.41]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[713.37, 794.96]]) <= 50.0"
] | ballsim | F_2_955 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
729,
704
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-575.44,
248.03
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
982,
869
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
690,
504
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 955,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 7508907
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (560, 932); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (826, 489); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (775, 807), initial velocity (-107.03, 779.3) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[764.31, 884.93]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[809.69, 843.3]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[853.26, 777.56]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[896.66, 712.01]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[818.29, 705.1]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[739.93, 698.19]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[661.56, 691.28]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[625.45, 684.37]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[703.82, 677.47]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[667.61, 736.7]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[650.43, 678.55]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[688.26, 664.35]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[698.54, 729.99]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[712.29, 700.77]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[736.46, 741.81]]) <= 50.0"
] | ballsim | F_2_509 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
775,
807
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-107.03,
779.3
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
560,
932
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
826,
489
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 509,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 9550278
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (999, 701); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (490, 734); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (705, 759), initial velocity (-471.49, 573.97) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[690.51, 832.13]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[675.03, 874.45]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[702.12, 822.12]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[768.58, 788.92]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[751.57, 755.72]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[738.96, 722.51]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[781.19, 689.31]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[714.73, 656.11]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[675.12, 610.78]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[696.06, 680.81]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[743.2, 738.21]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[790.34, 795.6]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[837.37, 852.86]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[788.06, 854.57]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[714.77, 842.44]]) <= 50.0"
] | ballsim | F_2_732 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
705,
759
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-471.49,
573.97
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
999,
701
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
490,
734
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 732,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6163974
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (526, 618); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (783, 999); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (741, 749), initial velocity (-537.75, -467.97) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[724.96, 720.37]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[795.07, 733.24]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[865.37, 746.14]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[895.17, 739.59]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[841.39, 692.79]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[787.6, 645.99]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[744.55, 603.66]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[813.95, 616.87]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[840.24, 650.84]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[806.58, 713.68]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[772.91, 776.52]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[765.28, 725.36]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[762.23, 654.33]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[751.28, 617.69]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[722.94, 683.1]]) <= 50.0"
] | ballsim | F_2_405 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
741,
749
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-537.75,
-467.97
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
526,
618
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
783,
999
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 405,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 2918789
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (821, 1016); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (920, 562); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (766, 753), initial velocity (433.57, -486.49) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[778.07, 743.58]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[740.28, 796.68]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[700.44, 847.48]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[663.47, 859.1]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[715.23, 819.5]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[766.99, 779.9]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[815.22, 744.73]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[788.12, 795.79]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[761.02, 736.51]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[733.92, 677.24]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[671.04, 662.83]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[660.86, 680.57]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[704.11, 729.1]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[747.47, 777.76]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[790.83, 773.38]]) <= 50.0"
] | ballsim | F_2_756 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
766,
753
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
433.57,
-486.49
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
821,
1016
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
920,
562
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 756,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 7500004
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (517, 857); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (503, 829); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (712, 745), initial velocity (513.98, 582.77) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[763.39, 803.28]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[814.78, 861.55]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[846.47, 833.05]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[867.49, 758.24]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[879.46, 687.79]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[807.88, 657.59]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[736.3, 627.39]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[697.92, 638.83]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[711.45, 715.35]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[724.98, 791.86]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[753.09, 861.35]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[790.92, 860.18]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[811.89, 785.57]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[832.91, 710.76]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[853.93, 635.96]]) <= 50.0"
] | ballsim | F_2_718 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
712,
745
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
513.98,
582.77
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
517,
857
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
503,
829
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 718,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4254953
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (1014, 734); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (972, 901); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (751, 752), initial velocity (-455.95, 487.65) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[705.39, 800.75]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[659.79, 849.51]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[679.07, 816.91]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[716.47, 761.61]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[753.87, 706.31]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[791.28, 651.01]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[813.53, 630.13]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[755.84, 663.72]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[698.15, 697.32]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[640.46, 730.91]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[619.93, 764.51]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[677.62, 798.1]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[735.17, 831.61]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[751.69, 841.43]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[693.75, 808.28]]) <= 50.0"
] | ballsim | F_2_396 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
751,
752
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-455.95,
487.65
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
1014,
734
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
972,
901
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 396,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6058017
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (1019, 752); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (594, 965); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (705, 750), initial velocity (494.14, -464.04) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[754.41, 703.59]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[803.83, 657.18]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[840.18, 623.81]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[774.83, 625.86]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[707.74, 635.55]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[645.72, 651.6]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[621.33, 714.85]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[611.7, 762.16]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[636.09, 698.9]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[660.48, 635.64]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[726.58, 624.71]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[793.67, 615.02]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[833.27, 618.5]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[783.86, 664.91]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[734.44, 711.32]]) <= 50.0"
] | ballsim | F_2_829 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
705,
750
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
494.14,
-464.04
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
1019,
752
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
594,
965
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 829,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 8920641
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (584, 527); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (999, 664); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (706, 705), initial velocity (354.01, -603.2) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[764.85, 747.45]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[822.9, 786.44]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[860.31, 837.54]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[803.73, 871.91]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[740.93, 883.08]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[678.14, 852.29]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[631.77, 812.02]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[636.5, 742.25]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[663.41, 769.59]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[697.95, 830.4]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[732.5, 891.22]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[767.05, 833.74]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[801.51, 773.07]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[784.75, 741.88]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[714.8, 741.39]]) <= 50.0"
] | ballsim | F_2_403 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
706,
705
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
354.01,
-603.2
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
584,
527
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
999,
664
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 403,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6022024
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (992, 679); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (476, 734); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (715, 708), initial velocity (-102.48, 632.84) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[704.75, 771.28]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[694.49, 834.56]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[697.58, 881.25]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[757.0, 857.17]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[816.43, 833.1]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[836.83, 857.95]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[810.13, 849.16]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[773.43, 796.59]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[736.74, 744.03]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[700.04, 691.47]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[663.35, 638.9]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[700.04, 678.34]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[743.03, 725.69]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[786.12, 773.15]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[829.21, 820.61]]) <= 50.0"
] | ballsim | F_2_122 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
715,
708
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-102.48,
632.84
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
992,
679
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
476,
734
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 122,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4795555
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (893, 520); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (902, 965); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (770, 734), initial velocity (-586.09, 469.06) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[711.38, 780.9]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[652.76, 827.8]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[695.42, 816.27]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[765.35, 788.98]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[770.55, 724.31]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[731.66, 666.01]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[673.04, 619.12]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[731.09, 639.56]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[725.28, 710.58]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[713.95, 784.8]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[702.62, 859.02]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[691.3, 845.99]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[680.0, 771.96]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[668.67, 697.74]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[665.19, 628.06]]) <= 50.0"
] | ballsim | F_2_626 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
770,
734
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-586.09,
469.06
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
893,
520
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
902,
965
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 626,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4541092
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (499, 637); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (887, 517); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (755, 779), initial velocity (-626.17, -412.86) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[692.38, 737.72]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[736.31, 747.74]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[807.8, 771.02]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[879.11, 794.24]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[838.85, 753.04]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[783.09, 720.83]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[727.33, 770.98]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[671.57, 821.13]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[674.64, 837.31]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[745.96, 814.09]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[817.27, 790.87]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[888.58, 767.65]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[887.84, 714.03]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[832.08, 759.85]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[776.32, 810.0]]) <= 50.0"
] | ballsim | F_2_628 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
755,
779
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-626.17,
-412.86
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
499,
637
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
887,
517
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 628,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 2471701
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (987, 626); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (528, 871); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (721, 734), initial velocity (64.64, -715.57) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[727.47, 662.44]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[733.94, 623.35]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[740.43, 695.08]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[746.9, 766.64]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[753.36, 838.2]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[759.83, 876.01]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[766.3, 804.45]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[772.77, 732.9]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[779.24, 661.34]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[718.76, 628.44]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[666.82, 623.63]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[673.29, 695.19]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[721.11, 742.68]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[779.85, 784.06]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[838.59, 825.45]]) <= 50.0"
] | ballsim | F_2_755 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
721,
734
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
64.64,
-715.57
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
987,
626
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
528,
871
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 755,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 1702205
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (934, 571); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (635, 976); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (743, 803), initial velocity (26.31, -729.86) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[748.49, 717.5]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[708.46, 697.99]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[637.71, 716.84]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[614.16, 758.06]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[661.86, 702.76]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[709.57, 647.46]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[699.33, 628.6]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[651.63, 683.9]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[603.92, 739.2]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[644.77, 743.37]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[716.52, 729.7]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[786.46, 718.3]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[789.09, 791.12]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[820.46, 850.27]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[806.87, 854.05]]) <= 50.0"
] | ballsim | F_2_18 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
743,
803
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
26.31,
-729.86
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
934,
571
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
635,
976
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 18,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 1813840
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (999, 649); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (539, 568); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (748, 742), initial velocity (-694.69, -363.36) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[715.02, 723.24]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[786.75, 754.91]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[847.49, 793.1]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[839.04, 871.05]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[802.43, 825.59]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[761.0, 759.04]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[719.57, 692.49]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[777.6, 673.84]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[743.5, 729.38]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[696.67, 792.27]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[649.84, 855.17]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[695.76, 801.75]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[746.51, 742.25]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[788.03, 688.01]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[710.93, 702.25]]) <= 50.0"
] | ballsim | F_2_636 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
748,
742
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-694.69,
-363.36
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
999,
649
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
539,
568
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 636,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 2571528
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (975, 867); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (789, 500); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (692, 792), initial velocity (-51.43, 610.8) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[686.85, 853.08]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[681.7, 871.6]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[676.53, 810.36]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[671.38, 749.28]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[666.23, 688.2]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[653.39, 663.73]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[655.93, 720.37]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[650.78, 781.46]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[645.63, 842.54]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[690.32, 874.84]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[745.78, 884.85]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[717.23, 878.5]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[667.04, 843.6]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[619.91, 806.77]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[614.76, 745.68]]) <= 50.0"
] | ballsim | F_2_1068 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
692,
792
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-51.43,
610.8
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
975,
867
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
789,
500
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1068,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 2933661
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (1006, 779); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (856, 1009); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (790, 742), initial velocity (642.37, 461.88) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[746.37, 788.19]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[682.14, 834.38]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[681.83, 843.68]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[740.86, 803.58]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[732.97, 724.85]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[725.09, 646.11]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[717.2, 646.85]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[709.32, 725.59]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[701.43, 804.32]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[693.54, 883.06]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[685.66, 823.97]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[677.77, 745.24]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[669.91, 666.7]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[686.83, 611.94]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[758.95, 644.49]]) <= 50.0"
] | ballsim | F_2_762 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
790,
742
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
642.37,
461.88
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1006,
779
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
856,
1009
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 762,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 3708833
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (475, 750); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (527, 590); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (783, 807), initial velocity (494.87, -348.29) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[832.49, 772.16]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[881.97, 737.32]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[867.43, 733.29]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[809.35, 750.26]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[751.27, 767.23]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[693.19, 784.2]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[694.96, 835.73]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[709.29, 894.52]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[752.38, 891.31]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[806.51, 864.24]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[860.63, 837.16]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[868.14, 787.66]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[855.57, 728.63]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[842.98, 669.45]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[827.95, 620.93]]) <= 50.0"
] | ballsim | F_2_382 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
783,
807
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
494.87,
-348.29
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
475,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
527,
590
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 382,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5156264
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (516, 875); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (1003, 737); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (716, 729), initial velocity (-123.61, 662.69) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[703.65, 795.26]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[761.03, 821.27]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[797.5, 856.89]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[765.04, 875.13]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[718.29, 826.87]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[700.67, 761.8]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[683.04, 696.74]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[665.41, 631.68]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[720.69, 658.02]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[780.21, 689.69]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[754.87, 721.35]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[695.35, 753.02]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[722.06, 734.9]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[779.23, 699.18]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[758.19, 663.46]]) <= 50.0"
] | ballsim | F_2_443 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
716,
729
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-123.61,
662.69
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
516,
875
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1003,
737
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 443,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 9928143
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (744, 479); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (801, 1020); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (783, 703), initial velocity (333.24, 575.97) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[816.33, 760.59]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[849.65, 818.19]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[883.06, 767.81]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[876.47, 733.26]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[809.91, 733.31]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[743.36, 733.36]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[676.81, 733.4]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[610.25, 733.45]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[626.77, 781.47]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[660.0, 839.13]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[664.85, 829.44]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[698.18, 771.84]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[731.42, 714.4]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[777.08, 710.89]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[832.09, 748.31]]) <= 50.0"
] | ballsim | F_2_1089 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
783,
703
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
333.24,
575.97
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
744,
479
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
801,
1020
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1089,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 8763120
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (576, 950); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (572, 930); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (790, 806), initial velocity (-552.87, -234.1) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[734.7, 782.59]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[714.69, 738.8]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[722.08, 679.06]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[729.46, 619.46]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[736.83, 660.9]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[744.2, 720.49]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[751.58, 780.09]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[758.95, 839.68]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[792.58, 884.12]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[840.5, 858.94]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[861.89, 807.43]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[869.26, 747.84]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[876.62, 688.39]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[822.15, 664.5]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[766.86, 641.09]]) <= 50.0"
] | ballsim | F_2_1038 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
790,
806
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-552.87,
-234.1
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
576,
950
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
572,
930
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1038,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4182114
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (477, 779); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (840, 1012); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (749, 804), initial velocity (-294.15, -582.62) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[719.58, 745.75]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[690.16, 687.5]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[665.92, 635.68]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[716.16, 677.33]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[766.4, 718.98]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[816.65, 760.63]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[866.89, 802.28]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[869.31, 770.38]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[842.08, 711.05]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[814.86, 651.73]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[780.96, 621.69]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[730.72, 663.34]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[680.6, 704.88]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[721.3, 702.72]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[785.19, 689.39]]) <= 50.0"
] | ballsim | F_2_34 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
749,
804
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-294.15,
-582.62
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
477,
779
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
840,
1012
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 34,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4200156
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (518, 601); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (934, 555); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (729, 723), initial velocity (577.43, 233.24) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[786.74, 746.32]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[844.48, 769.63]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[880.69, 780.49]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[831.61, 742.14]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[782.54, 703.78]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[733.47, 665.43]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[733.46, 679.85]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[715.68, 739.54]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[697.91, 799.24]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[680.14, 858.93]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[662.36, 860.61]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[644.59, 800.92]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[679.53, 807.41]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[733.78, 838.0]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[788.03, 868.59]]) <= 50.0"
] | ballsim | F_2_13 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
729,
723
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
577.43,
233.24
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
518,
601
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
934,
555
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 13,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 2593405
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (506, 645); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (496, 832); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (745, 730), initial velocity (349.06, -501.28) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[779.91, 679.88]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[814.82, 629.76]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[831.38, 660.27]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[841.07, 720.57]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[850.76, 780.87]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[860.46, 841.17]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[824.51, 879.51]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[765.41, 883.68]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[704.44, 879.73]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[725.31, 899.31]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[767.3, 854.95]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[809.29, 810.59]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[851.18, 766.34]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[893.17, 721.98]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[845.82, 720.65]]) <= 50.0"
] | ballsim | F_2_99 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
745,
730
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
349.06,
-501.28
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
506,
645
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
496,
832
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 99,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 7773913
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (513, 829); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (547, 906); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (709, 744), initial velocity (-492.89, -468.45) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[662.98, 695.26]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[658.65, 635.84]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[724.02, 616.53]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[789.23, 616.97]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[846.73, 640.68]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[830.81, 706.79]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[814.9, 772.91]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[798.98, 839.02]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[783.06, 880.64]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[767.14, 814.52]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[751.22, 748.41]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[735.31, 682.3]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[719.43, 616.35]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[703.51, 664.0]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[687.59, 730.11]]) <= 50.0"
] | ballsim | F_2_1050 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
709,
744
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-492.89,
-468.45
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
513,
829
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
547,
906
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1050,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 2921820
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (1025, 802); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (1025, 761); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (760, 794), initial velocity (579.58, 298.65) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[817.96, 823.86]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[762.68, 853.72]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[704.58, 883.65]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[660.09, 864.48]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[663.21, 799.35]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[666.34, 734.21]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[669.46, 669.07]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[672.59, 610.3]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[675.71, 675.43]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[678.84, 740.57]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[681.96, 805.71]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[685.09, 870.84]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[688.21, 849.95]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[691.33, 784.81]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[694.46, 719.68]]) <= 50.0"
] | ballsim | F_2_905 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
760,
794
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
579.58,
298.65
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1025,
802
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
1025,
761
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 905,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6995157
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (812, 1015); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (828, 988); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (743, 797), initial velocity (-633.33, 192.55) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[677.61, 813.68]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[615.3, 831.66]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[620.0, 765.45]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[624.68, 699.41]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[651.49, 661.12]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[716.89, 671.25]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[782.3, 681.38]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[847.7, 691.52]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[878.36, 718.39]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[845.49, 775.86]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[838.58, 719.6]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[833.89, 653.56]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[817.11, 640.77]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[784.24, 698.24]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[751.38, 755.71]]) <= 50.0"
] | ballsim | F_2_884 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
743,
797
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-633.33,
192.55
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
812,
1015
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
828,
988
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 884,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4328482
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (473, 785); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (552, 560); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (804, 708), initial velocity (556.78, 272.96) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[859.69, 735.29]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[903.89, 755.96]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[852.27, 721.3]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[800.78, 686.73]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[749.29, 652.16]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[766.74, 657.38]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[822.43, 684.68]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[878.12, 711.97]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[896.29, 760.93]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[863.39, 806.23]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[811.9, 840.8]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[760.41, 875.37]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[709.05, 875.92]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[679.18, 851.76]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[738.31, 870.46]]) <= 50.0"
] | ballsim | F_2_590 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
804,
708
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
556.78,
272.96
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
473,
785
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
552,
560
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 590,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 8743754
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (484, 687); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (715, 483); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (762, 726), initial velocity (520.16, -506.94) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[814.03, 686.95]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[866.05, 737.63]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[882.92, 771.46]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[810.85, 762.4]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[738.78, 753.34]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[695.95, 758.36]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[747.98, 809.04]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[800.01, 859.73]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[838.48, 851.05]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[863.36, 782.81]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[888.24, 714.58]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[835.1, 683.9]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[827.2, 737.08]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[834.41, 809.35]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[841.59, 881.52]]) <= 50.0"
] | ballsim | F_2_150 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
762,
726
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
520.16,
-506.94
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
484,
687
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
715,
483
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 150,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6931149
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (845, 995); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (697, 1001); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (803, 723), initial velocity (354.96, 495.09) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[838.5, 772.51]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[874.0, 771.71]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[890.14, 733.3]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[829.52, 739.28]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[768.9, 745.26]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[708.28, 751.25]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[647.66, 757.23]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[602.25, 754.43]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[637.75, 704.91]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[673.25, 655.4]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[708.75, 614.88]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[744.25, 664.39]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[779.65, 713.77]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[815.15, 763.29]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[850.65, 780.94]]) <= 50.0"
] | ballsim | F_2_1039 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
803,
723
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
354.96,
495.09
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
845,
995
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
697,
1001
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1039,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4763912
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (497, 773); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (880, 502); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (726, 764), initial velocity (427.88, 476.93) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[768.8, 811.71]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[811.6, 859.41]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[805.7, 879.06]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[743.01, 865.85]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[691.1, 857.83]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[740.51, 880.63]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[789.93, 839.86]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[839.34, 799.09]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[888.75, 758.31]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[871.91, 755.79]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[811.88, 778.2]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[751.84, 800.62]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[707.03, 830.23]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[726.93, 888.09]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[746.83, 827.18]]) <= 50.0"
] | ballsim | F_2_835 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
726,
764
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
427.88,
476.93
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
497,
773
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
880,
502
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 835,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6361120
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (730, 1014); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (837, 1010); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (745, 738), initial velocity (223.03, -579.59) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[767.31, 680.04]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[789.63, 622.08]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[812.0, 650.25]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[834.31, 708.21]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[856.63, 766.17]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[885.09, 797.17]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[903.32, 739.03]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[851.85, 716.49]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[792.38, 698.57]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[732.9, 680.65]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[673.43, 662.73]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[661.63, 672.34]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[706.75, 714.79]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[751.99, 757.34]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[797.23, 787.9]]) <= 50.0"
] | ballsim | F_2_80 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
745,
738
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
223.03,
-579.59
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
730,
1014
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
837,
1010
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 80,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 2691631
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (729, 1018); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (476, 736); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (794, 775), initial velocity (248.6, 559.27) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[819.74, 827.04]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[866.4, 787.44]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[913.17, 747.75]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[856.33, 767.9]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[798.72, 788.51]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[741.1, 809.11]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[697.25, 769.38]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[733.21, 771.38]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[792.44, 786.81]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[851.67, 802.24]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[858.44, 787.38]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[815.47, 743.8]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[772.61, 700.33]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[729.64, 656.75]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[686.67, 613.17]]) <= 50.0"
] | ballsim | F_2_161 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
794,
775
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
248.6,
559.27
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
729,
1018
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
476,
736
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 161,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5543182
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (510, 835); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (842, 996); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (800, 753), initial velocity (451.9, -535.88) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[845.19, 699.41]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[837.08, 676.61]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[767.91, 688.99]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[698.92, 701.34]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[705.69, 677.19]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[758.35, 630.93]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[811.01, 629.57]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[863.67, 675.83]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[890.14, 737.22]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[872.09, 787.62]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[805.7, 792.17]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[739.32, 769.68]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[705.0, 731.89]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[728.81, 665.97]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[752.61, 614.18]]) <= 50.0"
] | ballsim | F_2_775 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
800,
753
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
451.9,
-535.88
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
510,
835
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
842,
996
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 775,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 1597638
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (824, 499); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (554, 565); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (716, 783), initial velocity (270.02, -572.92) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[743.0, 725.7]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[721.05, 729.77]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[671.08, 768.93]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[621.22, 808.0]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[675.9, 786.72]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[734.66, 763.09]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[793.43, 739.46]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[852.19, 715.83]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[866.94, 717.6]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[817.09, 756.66]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[767.23, 795.72]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[717.38, 834.79]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[667.65, 873.75]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[709.63, 859.81]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[768.39, 836.18]]) <= 50.0"
] | ballsim | F_2_864 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
716,
783
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
270.02,
-572.92
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
824,
499
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
554,
565
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 864,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 2987454
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (946, 949); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (683, 499); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (698, 780), initial velocity (377.77, 587.96) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[735.77, 838.79]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[735.88, 875.83]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[665.9, 872.5]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[689.51, 815.26]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[721.54, 753.15]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[754.8, 696.44]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[810.61, 738.51]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[852.42, 719.23]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[846.45, 679.06]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[808.42, 715.58]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[796.11, 784.38]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[769.2, 844.74]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[715.91, 889.57]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[662.49, 844.52]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[622.64, 791.64]]) <= 50.0"
] | ballsim | F_2_760 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
698,
780
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
377.77,
587.96
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
946,
949
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
683,
499
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 760,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4375135
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (983, 874); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (583, 538); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (753, 735), initial velocity (575.18, -323.85) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[810.52, 702.63]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[868.04, 670.25]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[820.58, 688.42]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[759.4, 713.2]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[698.22, 737.98]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[637.04, 762.76]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[632.79, 787.54]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[693.97, 812.32]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[755.15, 837.1]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[738.27, 861.88]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[684.93, 876.82]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[695.48, 811.66]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[706.0, 746.66]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[729.47, 688.97]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[780.64, 647.27]]) <= 50.0"
] | ballsim | F_2_194 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
753,
735
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
575.18,
-323.85
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
983,
874
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
583,
538
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 194,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 2069560
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (562, 551); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (521, 880); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (767, 712), initial velocity (668.01, 275.72) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[833.8, 739.56]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[900.59, 767.13]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[849.94, 726.85]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[792.66, 682.78]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[736.86, 639.57]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[803.66, 667.13]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[870.45, 694.7]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[879.84, 755.41]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[870.32, 827.04]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[813.33, 871.27]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[756.06, 870.43]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[723.17, 812.28]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[713.68, 740.83]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[726.1, 681.86]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[792.89, 654.3]]) <= 50.0"
] | ballsim | F_2_890 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
767,
712
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
668.01,
275.72
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
562,
551
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
521,
880
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 890,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 3860230
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (954, 580); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (933, 554); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (803, 710), initial velocity (635.59, 130.82) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[780.89, 773.51]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[760.43, 835.11]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[739.92, 896.86]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[698.21, 865.29]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[653.07, 818.69]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[607.93, 772.08]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[639.91, 725.47]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[685.05, 678.87]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[724.41, 632.26]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[709.12, 623.1]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[744.53, 677.49]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[779.93, 731.89]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[815.24, 786.15]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[850.64, 840.54]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[819.72, 862.99]]) <= 50.0"
] | ballsim | F_2_573 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
803,
710
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
635.59,
130.82
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
954,
580
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
933,
554
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 573,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5867001
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (980, 892); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (494, 702); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (773, 780), initial velocity (201.5, 629.62) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[734.74, 814.84]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[672.94, 838.35]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[671.14, 886.15]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[715.6, 844.16]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[760.06, 795.23]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[804.51, 746.3]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[848.97, 697.38]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[834.46, 682.5]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[769.86, 696.54]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[726.82, 710.58]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[791.42, 724.62]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[845.42, 725.35]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[846.12, 659.4]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[809.99, 614.56]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[752.4, 638.65]]) <= 50.0"
] | ballsim | F_2_517 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
773,
780
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
201.5,
629.62
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
980,
892
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
494,
702
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 517,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5858447
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (890, 963); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (829, 499); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (733, 786), initial velocity (-676.1, -115.27) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[665.4, 774.48]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[604.9, 762.95]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[672.68, 751.4]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[740.28, 739.88]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[807.88, 728.35]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[875.48, 716.83]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[862.52, 744.11]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[811.35, 739.95]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[760.18, 700.0]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[709.0, 745.67]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[657.83, 791.35]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[617.59, 823.33]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[650.63, 763.42]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[683.76, 703.36]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[669.13, 670.88]]) <= 50.0"
] | ballsim | F_2_5 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
733,
786
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-676.1,
-115.27
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
890,
963
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
829,
499
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 5,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 9870260
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (827, 994); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (567, 933); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (793, 743), initial velocity (532.84, -547.24) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[846.3, 688.29]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[821.86, 678.44]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[747.66, 697.26]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[673.63, 716.03]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[599.61, 734.81]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[638.4, 688.47]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[691.7, 633.76]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[744.99, 635.19]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[798.29, 689.9]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[851.58, 744.61]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[877.33, 783.42]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[803.3, 764.65]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[729.47, 745.92]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[655.44, 727.14]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[622.87, 732.29]]) <= 50.0"
] | ballsim | F_2_201 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
793,
743
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
532.84,
-547.24
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
827,
994
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
567,
933
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 201,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6412071
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (671, 497); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (707, 1005); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (691, 734), initial velocity (-4.9, -707.89) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[708.01, 739.87]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[738.28, 803.86]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[768.63, 745.73]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[798.91, 681.74]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[865.35, 663.11]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[822.97, 701.54]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[767.94, 746.07]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[712.91, 790.6]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[657.88, 778.58]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[602.85, 734.05]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[641.2, 749.49]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[671.47, 800.26]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[701.67, 736.43]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[741.33, 713.58]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[796.36, 758.12]]) <= 50.0"
] | ballsim | F_2_120 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
691,
734
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-4.9,
-707.89
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
671,
497
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
707,
1005
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 120,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 444388
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (693, 1013); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (480, 699); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (732, 741), initial velocity (-390.99, -460.92) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[711.16, 694.91]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[750.25, 648.81]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[789.44, 618.16]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[828.52, 664.26]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[867.61, 710.35]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[906.7, 756.44]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[850.71, 747.64]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[791.24, 736.83]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[731.77, 726.01]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[714.15, 739.36]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[753.24, 785.46]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[795.31, 818.48]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[850.4, 793.98]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[899.47, 765.87]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[884.24, 710.89]]) <= 50.0"
] | ballsim | F_2_709 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
732,
741
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-390.99,
-460.92
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
693,
1013
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
480,
699
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 709,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 8092117
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (754, 484); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (613, 510); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (722, 796), initial velocity (489.54, -556.62) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[770.95, 740.34]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[819.9, 684.67]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[886.28, 704.67]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[877.5, 770.68]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[857.55, 842.07]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[824.16, 854.57]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[775.21, 798.9]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[726.26, 743.24]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[677.31, 728.69]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[628.36, 784.35]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[624.2, 838.87]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[689.36, 874.22]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[750.89, 894.26]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[794.27, 834.16]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[837.66, 774.05]]) <= 50.0"
] | ballsim | F_2_504 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
722,
796
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
489.54,
-556.62
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
754,
484
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
613,
510
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 504,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 1835231
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (647, 1005); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (509, 829); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (787, 710), initial velocity (-524.32, 324.74) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[734.56, 742.47]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[719.96, 753.1]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[774.44, 723.85]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[828.79, 694.67]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[868.78, 672.41]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[812.1, 696.7]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[755.41, 720.99]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[698.72, 745.28]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[735.38, 715.69]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[784.77, 678.75]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[834.16, 641.82]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[821.13, 634.93]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[761.62, 650.47]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[701.95, 666.04]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[659.06, 671.93]]) <= 50.0"
] | ballsim | F_2_738 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
787,
710
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-524.32,
324.74
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
647,
1005
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
509,
829
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 738,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 3564209
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (740, 480); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (895, 956); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (766, 788), initial velocity (222.28, -707.08) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[788.22, 717.3]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[822.13, 697.82]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[872.97, 727.78]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[844.59, 719.38]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[772.42, 736.25]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[700.26, 753.12]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[628.09, 769.99]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[631.49, 743.24]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[675.0, 698.22]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[633.56, 759.69]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[643.08, 791.75]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[717.03, 786.6]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[785.89, 775.31]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[807.35, 704.36]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[844.41, 701.73]]) <= 50.0"
] | ballsim | F_2_262 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
766,
788
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
222.28,
-707.08
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
740,
480
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
895,
956
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 262,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5837589
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (718, 478); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (519, 878); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (704, 724), initial velocity (672.38, 156.48) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[771.24, 739.65]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[838.47, 755.3]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[898.0, 766.44]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[850.83, 716.02]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[803.66, 686.66]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[756.5, 737.07]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[709.33, 787.49]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[757.02, 783.12]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[824.26, 767.47]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[891.49, 751.82]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[877.78, 782.92]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[830.61, 833.34]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[783.56, 883.62]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[736.39, 845.19]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[706.16, 785.0]]) <= 50.0"
] | ballsim | F_2_187 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
704,
724
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
672.38,
156.48
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
718,
478
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
519,
878
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 187,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 8028417
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (738, 479); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (679, 507); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (769, 805), initial velocity (204.08, 566.53) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[789.41, 861.67]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[796.28, 859.05]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[790.07, 799.01]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[783.87, 739.12]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[777.67, 724.5]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[771.46, 784.38]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[765.26, 844.27]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[756.7, 893.83]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[725.14, 842.56]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[693.57, 791.29]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[662.0, 740.02]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[630.43, 714.98]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[609.7, 766.12]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[641.26, 817.39]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[672.83, 868.66]]) <= 50.0"
] | ballsim | F_2_364 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
769,
805
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
204.08,
566.53
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
738,
479
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
679,
507
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 364,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4406730
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (1023, 762); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (977, 884); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (782, 711), initial velocity (526.59, 369.24) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[791.85, 723.2]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[733.55, 696.05]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[675.1, 668.83]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[662.92, 668.31]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[715.58, 705.22]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[768.24, 742.14]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[782.32, 741.98]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[742.89, 691.18]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[703.46, 640.37]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[664.03, 624.67]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[629.75, 678.44]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[605.46, 737.99]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[627.94, 770.42]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[691.67, 779.16]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[755.39, 787.9]]) <= 50.0"
] | ballsim | F_2_571 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
782,
711
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
526.59,
369.24
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1023,
762
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
977,
884
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 571,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5983662
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (764, 1006); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (882, 525); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (708, 802), initial velocity (650.51, 198.79) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[699.73, 722.22]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[694.82, 654.38]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[646.8, 647.37]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[671.48, 710.75]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[696.17, 774.13]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[662.58, 764.45]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[606.28, 726.29]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[656.12, 749.39]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[717.33, 779.05]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[782.45, 791.56]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[850.47, 791.39]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[877.54, 767.59]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[858.78, 774.53]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[864.93, 793.93]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[890.88, 734.72]]) <= 50.0"
] | ballsim | F_2_331 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
708,
802
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
650.51,
198.79
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
764,
1006
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
882,
525
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 331,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 7438575
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (646, 508); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (573, 950); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (700, 692), initial velocity (-39.69, -672.37) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[728.73, 768.08]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[754.31, 830.38]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[779.96, 886.39]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[805.55, 824.09]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[831.14, 761.78]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[856.72, 699.48]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[830.25, 667.24]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[825.92, 688.31]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[874.55, 734.89]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[880.26, 756.69]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[815.58, 737.87]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[750.91, 719.05]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[690.52, 718.32]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[644.31, 746.34]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[640.4, 745.61]]) <= 50.0"
] | ballsim | F_2_154 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
700,
692
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-39.69,
-672.37
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
646,
508
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
573,
950
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 154,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 39395
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (1026, 713); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (964, 889); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (760, 741), initial velocity (-210.03, -612.01) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[739.0, 679.79]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[718.01, 618.59]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[696.96, 657.0]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[675.96, 718.21]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[654.97, 779.42]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[639.01, 837.71]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[702.51, 850.11]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[750.59, 862.51]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[687.09, 874.92]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[684.65, 852.07]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[727.16, 803.29]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[769.66, 754.52]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[812.06, 705.86]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[786.04, 657.08]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[743.53, 608.3]]) <= 50.0"
] | ballsim | F_2_92 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
760,
741
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-210.03,
-612.01
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1026,
713
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
964,
889
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 92,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 9511502
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (872, 970); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (477, 700); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (691, 765), initial velocity (584.0, 246.18) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[748.64, 789.17]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[698.12, 750.91]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[725.22, 757.43]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[778.96, 761.64]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[820.91, 714.13]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[862.85, 666.62]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[803.37, 677.65]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[741.26, 690.2]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[718.93, 702.75]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[781.04, 715.3]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[843.15, 727.85]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[905.25, 740.4]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[888.03, 745.09]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[867.84, 685.01]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[836.97, 631.1]]) <= 50.0"
] | ballsim | F_2_876 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
691,
765
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
584,
246.18
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
872,
970
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
477,
700
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 876,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5470992
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (495, 770); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (608, 514); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (692, 720), initial velocity (447.31, -430.76) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[744.24, 709.84]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[803.22, 729.25]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[862.35, 748.71]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[896.13, 753.56]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[849.84, 712.18]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[803.55, 670.8]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[790.01, 645.19]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[851.24, 655.59]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[843.49, 705.82]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[821.89, 764.04]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[800.28, 822.27]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[778.67, 880.5]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[757.12, 847.19]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[735.51, 788.96]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[713.91, 730.73]]) <= 50.0"
] | ballsim | F_2_766 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
692,
720
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
447.31,
-430.76
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
495,
770
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
608,
514
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 766,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 8050717
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (481, 677); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (686, 499); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (752, 711), initial velocity (534.92, 366.02) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[805.49, 747.6]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[858.98, 784.19]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[854.79, 787.5]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[796.34, 759.48]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[737.9, 731.45]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[683.93, 717.06]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[660.35, 777.44]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[636.77, 837.82]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[699.27, 848.49]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[763.36, 858.26]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[827.45, 868.03]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[816.04, 834.22]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[775.64, 783.73]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[735.14, 733.12]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[705.11, 728.39]]) <= 50.0"
] | ballsim | F_2_104 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
752,
711
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
534.92,
366.02
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
481,
677
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
686,
499
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 104,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 9014979
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (995, 677); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (573, 935); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (742, 795), initial velocity (-342.94, -709.62) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[707.7, 724.03]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[673.4, 653.06]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[703.58, 619.19]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[782.2, 613.4]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[717.79, 607.61]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[677.08, 634.3]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[711.38, 705.27]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[745.68, 776.24]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[779.99, 847.21]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[814.29, 867.58]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[822.95, 828.76]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[776.26, 870.83]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[852.92, 853.44]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[803.49, 817.3]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[746.88, 800.71]]) <= 50.0"
] | ballsim | F_2_165 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
742,
795
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-342.94,
-709.62
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
995,
677
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
573,
935
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 165,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4446838
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (987, 898); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (975, 908); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (726, 694), initial velocity (-572.24, 246.42) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[668.77, 718.63]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[611.55, 743.27]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[654.46, 767.96]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[711.69, 792.6]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[768.92, 817.23]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[736.46, 841.86]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[679.23, 866.5]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[677.03, 822.12]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[688.31, 760.84]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[699.59, 699.56]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[710.87, 638.28]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[715.28, 607.13]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[698.91, 667.09]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[682.51, 727.19]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[666.1, 787.3]]) <= 50.0"
] | ballsim | F_2_24 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
726,
694
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-572.24,
246.42
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
987,
898
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
975,
908
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 24,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5822522
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (682, 489); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (586, 966); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (787, 698), initial velocity (-292.33, -708.4) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[781.27, 730.14]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[785.67, 806.65]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[790.07, 883.36]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[794.47, 825.9]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[798.86, 749.38]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[803.25, 672.87]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[839.58, 736.23]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[876.74, 803.24]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[811.39, 811.09]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[738.04, 810.55]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[777.49, 744.85]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[816.94, 679.16]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[846.37, 640.69]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[883.86, 706.76]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[892.03, 765.45]]) <= 50.0"
] | ballsim | F_2_616 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
787,
698
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-292.33,
-708.4
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
682,
489
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
586,
966
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 616,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 8874945
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (597, 955); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (715, 497); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (750, 719), initial velocity (214.72, -627.54) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[787.74, 727.5]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[834.32, 774.72]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[869.57, 815.45]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[805.39, 798.72]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[741.21, 782.0]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[730.2, 734.58]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[760.17, 724.81]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[803.77, 774.79]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[847.37, 824.77]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[818.97, 833.16]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[753.88, 820.36]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[760.94, 765.93]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[782.35, 703.34]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[823.68, 727.57]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[870.26, 774.79]]) <= 50.0"
] | ballsim | F_2_108 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
750,
719
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
214.72,
-627.54
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
597,
955
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
715,
497
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 108,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 7100600
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (488, 711); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (1008, 715); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (771, 697), initial velocity (-99.69, 665.35) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[761.04, 763.53]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[751.08, 830.06]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[741.09, 889.02]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[731.13, 822.49]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[721.17, 755.96]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[711.21, 689.43]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[701.25, 622.9]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[691.29, 657.86]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[743.23, 694.57]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[801.46, 728.26]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[744.91, 761.96]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[686.69, 795.65]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[673.18, 850.72]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[683.14, 868.52]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[693.1, 801.99]]) <= 50.0"
] | ballsim | F_2_345 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
771,
697
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-99.69,
665.35
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
488,
711
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
1008,
715
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 345,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 1886934
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (981, 887); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (1022, 773); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (744, 788), initial velocity (-15.0, -689.35) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[742.51, 719.05]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[741.02, 650.11]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[739.53, 633.24]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[738.04, 702.19]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[736.55, 771.13]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[735.06, 840.08]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[733.57, 876.75]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[732.08, 807.8]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[730.59, 738.86]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[729.1, 669.91]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[727.61, 613.27]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[726.13, 682.21]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[724.64, 750.98]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[723.15, 819.93]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[721.66, 888.87]]) <= 50.0"
] | ballsim | F_2_67 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
744,
788
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-15,
-689.35
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
981,
887
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
1022,
773
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 67,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 3917729
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (953, 902); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (497, 655); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (724, 725), initial velocity (-474.83, 427.23) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[676.51, 767.72]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[658.44, 824.62]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[681.78, 854.5]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[743.11, 836.61]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[690.17, 818.71]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[666.1, 818.76]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[718.32, 855.55]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[724.06, 892.34]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[725.17, 862.26]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[749.42, 803.18]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[773.66, 744.1]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[797.9, 685.02]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[822.09, 626.08]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[822.8, 670.04]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[818.99, 733.81]]) <= 50.0"
] | ballsim | F_2_789 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
724,
725
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-474.83,
427.23
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
953,
902
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
497,
655
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 789,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6721072
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (687, 1018); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (490, 659); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (785, 750), initial velocity (557.89, -480.64) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[840.79, 701.93]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[851.31, 675.66]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[778.76, 689.34]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[706.4, 702.99]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[720.44, 758.25]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[754.89, 823.33]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[813.93, 780.66]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[873.21, 736.96]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[859.72, 728.29]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[788.6, 747.39]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[717.48, 766.48]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[658.85, 791.58]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[710.0, 763.35]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[765.79, 715.28]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[821.58, 667.21]]) <= 50.0"
] | ballsim | F_2_23 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
785,
750
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
557.89,
-480.64
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
687,
1018
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
490,
659
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 23,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 4782557
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (514, 612); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (1003, 823); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (804, 756), initial velocity (268.88, 653.66) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[752.96, 776.37]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[682.92, 785.77]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[643.52, 812.99]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[670.39, 878.34]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[697.27, 842.07]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[724.15, 776.71]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[751.03, 711.36]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[777.91, 646.0]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[804.79, 633.58]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[831.67, 698.94]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[788.53, 723.87]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[718.48, 733.27]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[671.07, 755.61]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[697.95, 820.96]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[724.83, 886.32]]) <= 50.0"
] | ballsim | F_2_149 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
804,
756
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
268.88,
653.66
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
514,
612
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1003,
823
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 149,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 9514412
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (743, 497); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (615, 986); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (792, 766), initial velocity (-94.05, -700.68) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[783.32, 699.07]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[805.24, 766.28]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[827.22, 833.66]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[817.25, 882.46]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[808.97, 861.93]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[856.21, 809.35]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[903.45, 756.76]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[857.45, 758.01]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[788.28, 772.64]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[719.11, 787.26]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[649.95, 780.39]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[600.78, 754.22]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[622.65, 687.18]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[606.68, 716.52]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[615.59, 758.44]]) <= 50.0"
] | ballsim | F_2_275 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
792,
766
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-94.05,
-700.68
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
743,
497
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
615,
986
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 275,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 3286021
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (714, 1013); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (561, 573); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (743, 732), initial velocity (-788.47, 87.46) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[708.5, 766.34]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[753.79, 780.12]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[814.17, 728.36]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[874.4, 676.72]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[805.07, 687.47]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[727.14, 702.37]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[736.32, 767.54]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[775.15, 786.56]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[831.16, 730.36]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[873.04, 680.96]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[794.18, 689.7]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[719.54, 700.87]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[751.32, 773.34]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[795.42, 792.4]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[855.65, 740.76]]) <= 50.0"
] | ballsim | F_2_353 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
743,
732
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-788.47,
87.46
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
714,
1013
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
561,
573
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 353,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 5228682
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (471, 736); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (782, 1021); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (744, 782), initial velocity (208.19, 591.08) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[764.83, 768.69]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[785.65, 709.58]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[806.53, 650.33]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[816.23, 639.96]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[809.34, 702.24]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[802.46, 764.52]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[800.92, 803.37]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[821.75, 744.27]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[842.57, 685.16]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[850.25, 632.38]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[791.04, 611.8]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[730.18, 598.54]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[704.31, 636.84]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[697.43, 699.12]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[719.29, 747.56]]) <= 50.0"
] | ballsim | F_2_390 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
744,
782
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
208.19,
591.08
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
471,
736
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
782,
1021
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 390,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 3090700
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (682, 991); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (907, 981); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (754, 802), initial velocity (-640.51, 344.92) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[712.43, 737.98]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[669.68, 679.11]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[683.42, 652.78]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[755.78, 660.37]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[828.15, 667.96]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[854.38, 702.18]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[824.77, 768.64]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[768.42, 798.48]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[710.21, 754.83]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[652.01, 711.18]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[658.86, 705.08]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[725.75, 733.64]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[792.48, 762.13]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[825.43, 748.14]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[812.47, 676.55]]) <= 50.0"
] | ballsim | F_2_786 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
754,
802
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-640.51,
344.92
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
682,
991
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
907,
981
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 786,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 886712
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (480, 676); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (742, 1009); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (714, 710), initial velocity (641.56, -219.57) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[778.16, 688.05]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[842.32, 666.1]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[827.44, 689.81]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[776.34, 734.39]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[725.25, 778.97]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[674.15, 762.24]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[709.91, 767.8]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[774.08, 789.75]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[838.24, 811.7]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[848.84, 802.73]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[797.74, 758.15]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[746.64, 713.57]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[715.65, 680.64]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[779.81, 702.59]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[843.97, 724.54]]) <= 50.0"
] | ballsim | F_2_722 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
714,
710
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
641.56,
-219.57
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
480,
676
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
742,
1009
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 722,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 2559122
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 7 sides, radius 195.00m, center at (565, 926); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (933, 569); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (802, 792), initial velocity (701.54, -178.64) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[862.95, 785.69]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[861.14, 858.05]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[801.12, 873.8]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[777.41, 833.56]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[807.19, 767.57]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[784.38, 767.51]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[745.11, 802.33]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[815.28, 784.46]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[858.55, 800.31]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[855.39, 872.02]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[788.01, 866.31]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[779.73, 818.65]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[809.44, 752.82]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[770.5, 772.98]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[758.05, 794.89]]) <= 50.0"
] | ballsim | F_2_212 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
802,
792
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
701.54,
-178.64
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
565,
926
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
933,
569
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 212,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 9363087
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (667, 993); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (494, 856); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (705, 795), initial velocity (69.03, -617.65) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[711.91, 733.23]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[718.82, 671.46]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[725.74, 611.22]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[732.65, 672.99]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[739.56, 734.76]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[746.47, 793.21]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[753.38, 731.44]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[760.29, 669.67]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[767.2, 612.86]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[774.11, 674.63]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[781.02, 736.39]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[787.93, 798.16]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[839.71, 803.47]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[878.15, 783.01]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[853.34, 726.03]]) <= 50.0"
] | ballsim | F_2_543 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
705,
795
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
69.03,
-617.65
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
667,
993
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
494,
856
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 543,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 754058
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (1020, 687); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (1017, 778); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (780, 713), initial velocity (529.2, 284.29) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[789.67, 741.42]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[736.74, 769.84]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[683.68, 798.32]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[630.75, 826.74]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[679.68, 796.37]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[730.75, 764.75]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[781.83, 733.13]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[788.6, 727.09]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[735.67, 755.51]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[682.74, 783.92]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[629.81, 812.34]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[667.68, 788.35]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[718.62, 756.81]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[769.7, 725.2]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[787.79, 712.62]]) <= 50.0"
] | ballsim | F_2_232 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
780,
713
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
529.2,
284.29
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1020,
687
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
1017,
778
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 232,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 3670373
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 7 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (950, 587); initial orientation 0.000 rad\n- Container 3: regular polygon with 7 sides, radius 195.00m, center at (483, 689); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (724, 766), initial velocity (518.45, 317.47) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[775.86, 797.74]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[827.71, 829.48]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[849.98, 846.98]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[792.83, 826.23]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[735.67, 805.48]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[678.52, 784.73]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[703.41, 803.48]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[755.27, 835.22]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[807.12, 866.96]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[849.09, 855.39]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[882.02, 804.29]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[867.69, 787.77]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[807.36, 793.85]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[746.89, 799.96]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[686.42, 806.06]]) <= 50.0"
] | ballsim | F_2_694 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
724,
766
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
518.45,
317.47
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
},
{
"center": [
950,
587
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
483,
689
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 7
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 694,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 737054
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (615, 514); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (490, 843); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 30.0m, initial position (720, 795), initial velocity (-374.71, -605.05) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[682.52, 734.5]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[664.02, 732.78]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[735.34, 730.57]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[806.49, 728.38]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[877.63, 726.18]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[875.56, 766.24]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[838.09, 826.74]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[800.61, 887.23]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[763.14, 838.04]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[725.66, 777.54]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[688.19, 717.04]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[676.33, 746.42]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[747.29, 744.23]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[818.44, 742.03]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[889.58, 739.83]]) <= 50.0"
] | ballsim | F_2_885 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
720,
795
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-374.71,
-605.05
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
615,
514
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
490,
843
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 885,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 6261710
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 195.00m, center at (750, 750); initial orientation 0.000 rad\n- Container 2: regular polygon with 6 sides, radius 195.00m, center at (513, 615); initial orientation 0.000 rad\n- Container 3: regular polygon with 6 sides, radius 195.00m, center at (993, 838); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (7 sides), radius 30.0m, initial position (705, 780), initial velocity (-568.48, 212.78) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[648.16, 801.29]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[651.52, 787.81]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[698.49, 749.12]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[745.34, 710.52]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[792.19, 671.92]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[839.04, 633.32]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[791.71, 649.12]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[734.87, 670.41]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[708.39, 709.22]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[718.38, 769.09]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[728.36, 828.95]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[738.35, 888.81]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[748.31, 830.71]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[758.3, 770.84]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[768.28, 710.98]]) <= 50.0"
] | ballsim | F_2_211 | 2 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
255,
200
],
"density": 1,
"position": [
705,
780
],
"radius": 30,
"restitution": 1,
"rotation_affected": false,
"sides": 7,
"velocity": [
-568.48,
212.78
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
513,
615
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
},
{
"center": [
993,
838
],
"diameter": 390,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "medium",
"difficulty_config": {
"description": "Medium: 标准空间，复杂布局，高密度"
},
"difficulty_level": 2,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 211,
"multiple_containers": {
"container_count": 3,
"container_layout": "non_overlapping",
"interaction_rule": "no_container_collisions"
},
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "F",
"seed": 404036
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
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