What mathematical puzzle that consists of three pegs and a number of disks of different sizes in which the objective is to move the entire stack of disks from one peg to another is used in computer science to teach recursion to students of programming?
"from a peg to another peg are symmetric with respect to the center points. 2)The smallest disk is the first and last disk to move. 3)Groups of the smallest disk moves alternate with single moves of other disks. 4)The number of disks moves specified by C(n) and A(n) are minimal. Although the three-peg version has a simple recursive solution as outlined above which has long been known, the ""optimal"" solution for the Tower of Hanoi problem with four pegs (called Reve's puzzle) was not verified until 2014, and in the case of more than four pegs a proof that the"
"3, and so on until only one disk is left. This is called recursion. This algorithm can be schematized as follows. Identify the disks in order of increasing size by the natural numbers from 0 up to but not including ""h"". Hence disk 0 is the smallest one, and disk ""h"" − 1 the largest one. The following is a procedure for moving a tower of ""h"" disks from a peg A onto a peg C, with B being the remaining third peg: By means of mathematical induction, it is easily proven that the above procedure requires the minimal number"
"when teaching programming. As in many mathematical puzzles, finding a solution is made easier by solving a slightly more general problem: how to move a tower of ""h"" (height) disks from a starting peg ""f"" = A (from) onto a destination peg ""t"" = C (to), B being the remaining third peg and assuming ""t"" ≠ ""f"". First, observe that the problem is symmetric for permutations of the names of the pegs (symmetric group ""S""). If a solution is known moving from peg A to peg C, then, by renaming the pegs, the same solution can be used for every"
"is always one possibility, except when all disks are on the same peg, but in that case either it is the smallest disk that must be moved or the objective has already been achieved. Luckily, there is a rule that does say where to move the smallest disk to. Let ""f"" be the starting peg, ""t"" the destination peg, and ""r"" the remaining third peg. If the number of disks is odd, the smallest disk cycles along the pegs in the order ""f"" → ""t"" → ""r"" → ""f"" → ""t"" → ""r"", etc. If the number of disks is"
"""auxiliary"" stacks are inverted. A = [3, 2, 1] B = [] C = [] def move(n, source, target, auxiliary): move(3, A, C, B) The list of moves for a tower being carried from one peg onto another one, as produced by the recursive algorithm, has many regularities. When counting the moves starting from 1, the ordinal of the disk to be moved during move ""m"" is the number of times ""m"" can be divided by 2. Hence every odd move involves the smallest disk. It can also be observed that the smallest disk traverses the pegs ""f"", ""t"", ""r"","
"or Blue. At the beginning of the puzzle the disks are stacked on the S post in order of decreasing size (i.e. the largest disk is at the bottom), and such that all disks have their Red side facing upwards. The objective of the puzzle (in its ""basic"" version) is to move the entire stack, one disk at a time, to the D post, maintaining the order from largest to smallest disk, but with the Blue sides facing upwards. The rules governing the movement of the disks are as follows: In order to illustrate the rules of the MToH, and"
"other choice of starting and destination peg. If there is only one disk (or even none at all), the problem is trivial. If ""h"" = 1, then simply move the disk from peg A to peg C. If ""h"" > 1, then somewhere along the sequence of moves, the largest disk must be moved from peg A to another peg, preferably to peg C. The only situation that allows this move is when all smaller ""h"" − 1 disks are on peg B. Hence, first all ""h"" − 1 smaller disks must go from A to B. Then move the"
"largest disk and finally move the ""h"" − 1 smaller disks from peg B to peg C. The presence of the largest disk does not impede any move of the ""h"" − 1 smaller disks and can be temporarily ignored. Now the problem is reduced to moving ""h"" − 1 disks from one peg to another one, first from A to B and subsequently from B to C, but the same method can be used both times by renaming the pegs. The same strategy can be used to reduce the ""h"" − 1 problem to ""h"" − 2, ""h"" −"
"three corner nodes, which have two: it is always possible to move the smallest disk to one of the two other pegs; and it is possible to move one disk between those two pegs ""except"" in the situation where all disks are stacked on one peg. The corner nodes represent the three cases where all the disks are stacked on one peg. The diagram for ""n"" + 1 disks is obtained by taking three copies of the ""n""-disk diagram—each one representing all the states and moves of the smaller disks for one particular position of the new largest disk—and joining"
"the basic principles of the original puzzle (i.e. three plates that the pancakes could be moved onto, not being able to put a larger pancake onto a smaller one, etc.) In the film ""Rise of the Planet of the Apes"" (2011), this puzzle, called in the film the ""Lucas Tower"", is used as a test to study the intelligence of apes. The puzzle is featured regularly in adventure and puzzle games. Since it is easy to implement, and easily recognised, it is well-suited to use as a puzzle in a larger graphical game (e.g. """" and ""Mass Effect""). Some implementations"
"even, this must be reversed: ""f"" → ""r"" → ""t"" → ""f"" → ""r"" → ""t"" etc. The position of the bit change in the Gray code solution gives the size of the disk moved at each step: 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, ... , a sequence also known as the ruler function, or one more than the power of 2 within the move number. In the Wolfram Language, codice_5 gives moves for the 8-disk puzzle. The game can be represented by an undirected graph, the nodes representing distributions of"
"where all disks are not necessarily on the same peg, and to arrive in a minimal number of moves at another given configuration. In general it can be quite difficult to compute a shortest sequence of moves to solve this problem. A solution was proposed by Andreas Hinz, and is based on the observation that in a shortest sequence of moves, the largest disk that needs to be moved (obviously one may ignore all of the largest disks that will occupy the same peg in both the initial and final configurations) will move either exactly once or exactly twice. The"
"that its point is to form a picture on-screen. The last square of the puzzle is then displayed automatically once the other pieces have been lined up. Sliding puzzle A sliding puzzle, sliding block puzzle, or sliding tile puzzle is a combination puzzle that challenges a player to slide (frequently flat) pieces along certain routes (usually on a board) to establish a certain end-configuration. The pieces to be moved may consist of simple shapes, or they may be imprinted with colors, patterns, sections of a larger picture (like a jigsaw puzzle), numbers, or letters. Sliding puzzles are essentially two-dimensional in"
"B – C – A and A – C – B – A respectively. The moving direction of the disk must be clockwise. It suffices to represent the sequence of disks to be moved. The solution can be found using two mutually recursive procedures: To move ""n"" disks counterclockwise to the neighbouring target peg: To move ""n"" disks clockwise to the neighbouring target peg: Let C(n) and A(n) represent moving n disks clockwise and counterclockwise, then we can write down both formulas: The solution for the Cyclic Hanoi has some interesting properties: 1)The move-patterns of transferring a tower of disks"
"over another piece's peg, the two pegged piece can be moved. Once a two peg piece has been slid to the side, the notched piece can be moved to its second notch and lifted off. Once the notch piece is removed, all the others become loose and the puzzle usually crumbles into its remaining pieces. The challenge is to put it back together. Equilibrium was sold in a cardboard box, with a solution sheet enclosed. The secret in putting it back together involves not only holding the pieces in such a way that all three angles are supported so they"
"2 according as whether the number of disks is even or odd. Another formulation is from peg codice_3 to peg codice_4. Furthermore, the disk to be moved is determined by the number of times the move count (""m"") can be divided by 2 (i.e. the number of zero bits at the right), counting the first move as 1 and identifying the disks by the numbers 0, 1, 2 etc. in order of increasing size. This permits a very fast non-recursive computer implementation to find the positions of the disks after m moves without reference to any previous move or distribution"
"possible to list a recursion relation for the total number of moves made by disk k, which we denote by formula_5 and formula_6 for the RBB and RRB algorithms respectively (note that formula_7 is independent of the total number of disks ""n"" in the puzzle). Again working through the algorithms, and using the equality formula_8, it is simple to show that From these recursive relations, it is quite simple to derive closed form expressions for formula_1 and formula_5, which are given by As can be seen, these quantities scale as 3, in contrast to the classical ToH puzzle which scales"
"Combination puzzle A combination puzzle, also known as a sequential move puzzle, is a puzzle which consists of a set of pieces which can be manipulated into different combinations by a group of operations. The puzzle is solved by achieving a particular combination starting from a random (scrambled) combination. Often, the solution is required to be some recognisable pattern such as 'all like colours together' or 'all numbers in order'. The most famous of these puzzles is the original Rubik's Cube, a cubic puzzle in which each of the six faces can be independently rotated. Each of the six faces"
"Sliding puzzle A sliding puzzle, sliding block puzzle, or sliding tile puzzle is a combination puzzle that challenges a player to slide (frequently flat) pieces along certain routes (usually on a board) to establish a certain end-configuration. The pieces to be moved may consist of simple shapes, or they may be imprinted with colors, patterns, sections of a larger picture (like a jigsaw puzzle), numbers, or letters. Sliding puzzles are essentially two-dimensional in nature, even if the sliding is facilitated by mechanically interlinked pieces (like partially encaged marbles) or three-dimensional tokens. As this example shows, some sliding puzzles are mechanical"
"the graph is: The sides of the outermost triangle represent the shortest ways of moving a tower from one peg to another one. The edge in the middle of the sides of the largest triangle represents a move of the largest disk. The edge in the middle of the sides of each next smaller triangle represents a move of each next smaller disk. The sides of the smallest triangles represent moves of the smallest disk. In general, for a puzzle with ""n"" disks, there are 3 nodes in the graph; every node has three edges to other nodes, except the"
"opposite poles attracting one another. Magnets inside each disk physically prevent illegal moves. One of the striking features of the classical ToH puzzle is its relation to the base 2: the minimum number of total moves required to solve the puzzle is 2 − 1 (where ""n"" is the number of disks), while the minimum number of moves made by disk ""k"" is 2 (disks are numbered bottom up so that ""k"" = 1 being the largest disk, and ""k"" = ""n"" being the smallest). It will be shown below that just as the original ToH puzzle is related to"
"3-colorability of a planar graph, which is known to be NP-complete, to a Corral Puzzle. Bag (puzzle) Bag (also called Corral or Cave) is a binary-determination logic puzzle published by Nikoli. Bag is played on a rectangular grid, usually of dashed lines, in which numbers appear in some of the cells. The object is to draw a single, continuous loop along the lines of the grid, which contains all the numbers on the grid. Additionally, each number denotes the sum of all cells visible in any orthogonal direction before line of the loop is reached. For example, a 2 cell"
"""f"", ""t"", ""r"", etc. for odd height of the tower and traverses the pegs ""f"", ""r"", ""t"", ""f"", ""r"", ""t"", etc. for even height of the tower. This provides the following algorithm, which is easier, carried out by hand, than the recursive algorithm. In alternate moves: For the very first move, the smallest disk goes to peg ""t"" if ""h"" is odd and to peg ""r"" if ""h"" is even. Also observe that: With this knowledge, a set of disks in the middle of an optimal solution can be recovered with no more state information than the positions of each"
"With these operations the combination of the parts can be changed. The goal of the game is to obtain a specific combination, such as forming a picture or ordering the number on the segments. Puck puzzle The Puck puzzle (also Picture Disc and Kép Korong in the original Hungarian) is a combination puzzle invented in 1980 by Hungarian physicist András Végh. The disc shaped puzzle consists of 2 central pieces - the inner disc - and segments forming a ring around the inner disc. The number of segments is an even number - most commonly six, eight or twelve. The"
"N-dimensional sequential move puzzle The Rubik's Cube is the original and best known of the three-dimensional sequential move puzzles. There have been many virtual implementations of this puzzle in software. It is a natural extension to create sequential move puzzles in more than three dimensions. Although no such puzzle could ever be physically constructed, the rules of how they operate are quite rigorously defined mathematically and are analogous to the rules found in three-dimensional geometry. Hence, they can be simulated by software. As with the mechanical sequential move puzzles, there are records for solvers, although not yet the same degree"
"method to optimize the back-up cycle. Every tape corresponds to a disk in the puzzle, and every disk movement to a different peg corresponds with a backup to that tape. So the first tape is used every other day (1, 3, 5, 7, 9, ...), the second tape is used every fourth day (2, 6, 10, ...), the third tape is used every eighth day (4, 12, 20, ...). A set of ""n"" tapes (or other media) will allow backups for 2 days before the last set is recycled. So, 3 tapes will give 4 days' worth of backups, and"
"Pick-up sticks Pick-up sticks or pick-a-stick is a game of physical and mental skill in which a bundle of ""sticks"", between 8 and 20 centimeters long, are dropped as a loose bunch onto a table top, jumbling into a random pile. Each player in turn tries to remove a stick from the pile without disturbing any of the others. The game is known by several names including jackstraws (or jack straws), spellicans, and spillikins, and appears in a line of a nursery rhyme: ""...five, six, pick-up sticks!"" The sticks may be made of almost any material, such as ivory, bone,"
"Tiling puzzle Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given shape first and then rearrange the pieces into another shape. Other tiling puzzles ask you to dissect a given shape while fulfilling certain conditions. The two latter types of tiling puzzles are also called dissection puzzles. Tiling puzzles may be made from wood, metal, cardboard, plastic or any other sheet-material. Many tiling puzzles are now available as computer games."
"a 2D computer screen is an example of a general type of an ""n""-dimensional puzzle projected on to a (""n"" – 2)-dimensional space. The 3D analogue of this is to project the cube on to a 1-dimensional representation, which is what Vanderschel's program is capable of doing. Vanderschel bewails that nobody has claimed to have solved the 1D projection of this puzzle. However, since records are not being kept for this puzzle it might not actually be the case that it is unsolved. N-dimensional sequential move puzzle The Rubik's Cube is the original and best known of the three-dimensional sequential"
"for teaching the concepts of good sportsmanship and the branch of artificial intelligence that deals with the searching of game trees. It is straightforward to write a computer program to play tic-tac-toe perfectly or to enumerate the 765 essentially different positions (the state space complexity) or the 26,830 possible games up to rotations and reflections (the game tree complexity) on this space. The game can be generalized to an m,n,k-game in which two players alternate placing stones of their own color on an ""m""×""n"" board, with the goal of getting ""k"" of their own color in a row. Tic-tac-toe is"
"have not (and often could not) be made by bandaging. Combination puzzle A combination puzzle, also known as a sequential move puzzle, is a puzzle which consists of a set of pieces which can be manipulated into different combinations by a group of operations. The puzzle is solved by achieving a particular combination starting from a random (scrambled) combination. Often, the solution is required to be some recognisable pattern such as 'all like colours together' or 'all numbers in order'. The most famous of these puzzles is the original Rubik's Cube, a cubic puzzle in which each of the six"
"Puck puzzle The Puck puzzle (also Picture Disc and Kép Korong in the original Hungarian) is a combination puzzle invented in 1980 by Hungarian physicist András Végh. The disc shaped puzzle consists of 2 central pieces - the inner disc - and segments forming a ring around the inner disc. The number of segments is an even number - most commonly six, eight or twelve. The outer ring can be rotated around the central disc, similar to a wheel being rotated around its axis. The two parts of the central disc can also be rotated, together with the attached ring."
"Survo puzzle A Survo puzzle is a kind of logic puzzle presented (in April 2006) and studied by Seppo Mustonen. The name of the puzzle is associated with Mustonen's Survo system, which is a general environment for statistical computing and related areas. In a Survo puzzle, the task is to fill an ""m"" × ""n"" table with integers 1, 2, ..., ""m""·""n"" so that each of these numbers appears only once and their row and column sums are equal to integers given on the bottom and the right side of the table. Often some of the integers are given readily"