"mantle length. Cephalopods of comparable size to the largest present day squid are known from fossil remains, including enormous examples of ammonoids, belemnoids, nautiloids, orthoceratoids, teuthids, and vampyromorphids (see Teichert & Kummel, 1960; Stevens, 1988; Eyden, 2003b; Larson, 2010; Iba ""et al."", 2015). What follows is an overview of the present state of scientific knowledge on the maximum size of the eight extant cephalopod species (six squids and two octopuses) that are known or estimated to reach at least in mass. Extinct taxa are considered separately at the end. Though a substantial number of colossal squid (""Mesonychoteuthis hamiltoni"") remains have"
"objects are the ""n"" element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for . Six similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others."
"24-dimensional unit balls that can simultaneously touch a single unit ball (compare with 6 in dimension 2, as the maximum number of pennies which can touch a central penny; see kissing number). This arrangement of 196,560 unit balls centred about another unit ball is so efficient that there is no room to move any of the balls; this configuration, together with its mirror-image, is the ""only"" 24-dimensional arrangement where 196,560 unit balls simultaneously touch another. This property is also true in 1, 2 and 8 dimensions, with 2, 6 and 240 unit balls, respectively, based on the integer lattice, hexagonal"
"by the players. At the beginning of every round each player may hold any quantity of coins, from zero to a maximum of three in their closed fist, extended into the circle of play. The coins may be of any denomination, and indeed the values of the coins are irrelevant: in fact, any suitable objects could be used in place of coins. In general, the number ""three"" can be replaced with any other integer ""n"" ≥ 1, but all other rules are the same, except now each player chooses to hold any number 0, 1, ..., ""n"" of coins in"
"Chink-a-chink Chink-a-chink is a simple close-up magic trick in which a variety of small objects, usually four, appear to magically transport themselves from location to location when covered by the performer's hands, until the items end up gathered together in the same place. Variations, especially the Sympathetic Coins, have been performed since the 1800s. Popular variations are Shadow Coins, Coins-n-Cards and Matrix. A variation using playing cards as the objects is known as Sympathetic Aces. In the typical layout, the magician places four small objects on a table in a square, rectangular or diamond formation (although even a single straight"
"""y"") and (""x"", ""y""), whereas that of a solution circle can be written as (""x"", ""y""). Similarly, the radii of the given circles and a solution circle can be written as ""r"", ""r"", ""r"" and ""r"", respectively. The requirement that a solution circle must exactly touch each of the three given circles can be expressed as three coupled quadratic equations for ""x"", ""y"" and ""r"": The three numbers ""s"", ""s"" and ""s"" on the right-hand side, called signs, may equal ±1, and specify whether the desired solution circle should touch the corresponding given circle internally (""s"" = 1) or externally"
"The precise figure is 4 400 411 583 858 825 100 777 127 453 704 140 502 784 413 155 112 522 644 357 120 000 000 (roughly 4.4 unvigintillion on the short scale or 4.4 undecillion on the long scale). Dogic The Dogic () is an icosahedron-shaped puzzle like the Rubik's Cube. The 5 triangles meeting at its tips may be rotated, or 5 entire faces (including the triangles) around the tip may be rotated. It has a total of 80 movable pieces to rearrange, compared to the 20 pieces in the Rubik's Cube. The Dogic was patented by"
"be in an even permutation exactly when the corners are. (When arrangements of centres are also permitted, as described below, the rule is that the combined arrangement of corners, edges, and centres must be an even permutation.) Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 2 (2,048) possibilities. which is approximately 43 quintillion. The puzzle was originally advertised as having ""over 3,000,000,000 (three billion) combinations but only one solution"". To put this into perspective, if one had as many standard sized Rubik's Cubes as there are permutations, one could cover"
"be the circle, whose centre is to be found. Let A be a point on (C). A circle (C1) centered at A meets (C) at B and B'. Two circles (C2) centered at B and B', with radius AB, cross again at point C. A circle (C3) centered at C with radius AC meets (C1) at D and D'. Two circles (C4) centered at D and D' with radius AD meet at A, and at O, the sought center of (C). Note: for this to work the radius of circle (C1) must be neither too small nor too large. More"
"circular. Alternatively, this can be seen from considering the link diagram: if one assumes that circles 1 and 2 touch at their two crossing points, then they either lie in a plane or a sphere. In either case, the third circle must pass through this plane or sphere four times, without lying in it, which is impossible; see . It is, however, true that one can use ellipses (right picture). These may be taken to be of arbitrarily small eccentricity; i.e. no matter how close to being circular their shape may be, as long as they are not perfectly circular,"
"proceeds clockwise around the circle until each player has ventured a guess regarding the total number of coins, and no player can guess the same total as any other player. The call of ""Spoof!"" is sometimes used to mean ""zero"". After all players have made their guess, they open their fists and display their coins for the group to count the total. It is illegal to open your hand without calling out your guess. The player who has correctly guessed the total number of coins is eliminated from the game and the remainder of the group proceeds to the next"
"symmetrical. Although each side had the same array of pieces, the numbers on them differed, allowing different possible captures and winning configurations to the two players. The rules below describe the most common version of the game, played through much of the Middle Ages and Renaissance. There was also a variant propounded by Fulke in the 16th century, with significantly different (and somewhat more consistent) capture rules. There are four types of pieces, which are Rounds, Triangles, Squares, and Pyramids. There were a variety of capture methods. Pieces do not land on another piece to capture it, but instead remain"
"resulting in the coin's features being presented right side up when the coin is turned on a horizontal axis. This is called a ""coin turn"" alignment. Correct alignment for a ""coin turn"" is exactly 180 degrees; however due to realities of machine process, this can vary by a fraction of a degree. In rare cases, this deviation can be up to several degrees, and very easily noticeable. However, even very small deviations can establish the production run in which a coin was minted. When this is done, it is called a die marriage. Die marriage is particularly important with regards"
"Circle, either until time runs out or until the remaining categories have been correctly guessed. If all six categories are guessed before time runs out, the contestant wins the top prize; if not, he/she wins money for the guessed categories. The clue-giver can pass on a category and then return to it after playing through all six, if time allows. The values for individual categories during standard gameplay are shown in the table below. Category numbering proceeds across the bottom row of the pyramid (left/center/right), then the middle (left/right), and finally the single one at the peak. On the 1970s"
"this forms a perfect hexagon with four rings on a side. The marbles go into a shared pool. The object is to capture four white, or five gray, or six black marbles, or three marbles of each color. (A quicker variant can be played in which the object is to capture three white marbles, or four grays, or five blacks, or two of each color. See below.) If any two marbles are adjacent on the board, and there is room for one to ""jump"" the other, landing on a ring immediately opposite the other, a player must jump instead of"
"in a square, one behind the other in groups of three. They are each worth one points except for the one at the center of the square which is worth two. The aim is to scatter the pins as far away from their original location as possible. There is also a smaller 20 cm tall pin called ""miko"" or ""cuatro"" (because it is worth four points). Depending on they type of game, each player has 2–4 balls per round and the ""miko"" is placed in two different positions behind the big pins. In this variant, the ball and the central"
"equal which means that E belongs to the radical axis. Any pair of circles has two centers of similarity, therefore, three circles would have six centers of similarity, two for each distinct pair of given circles. Remarkably, these six points lie on four lines, three points on each line. Here is one way to show this. Consider the ""plane"" of the three circles (Figure 9). Offset each center point perpendicularly to the plane by a distance equal to the corresponding radius. The centers can be offset to either side of the plane. The three offset points define a single plane."
"good approximation to the circumference. To compute ""u"" and ""U"" for large ""n"", Archimedes derived the following doubling formulae: Starting from a hexagon, Archimedes doubled ""n"" four times to get a 96-gon, which gave him a good approximation to the circumference of the circle. In modern notation, we can reproduce his computation (and go further) as follows. For a unit circle, an inscribed hexagon has ""u"" = 6, and a circumscribed hexagon has ""U"" = 4. Doubling seven times yields The last entry of the table has ⁄ as one of its best rational approximations; i.e., there is no better"
"three circles overlapping each other equally like in a triple Venn diagram with the overlapping parts erased. The diameter of the overlapping part is equal to half the radius of the three circles. Then three inner circles are drawn in with radius of the original circles so that it is tangent to the outside three overlapping circles. A tiny circle in center has a diameter of the radius of the three inner circles, and arcs are erased at 90°, 210°, and 330°. The arcs of the inner circles and the tiny circle are connected by a line. Finally, the ring"
"adhere to one another are called ""bonded pairs"". Softer objects, such as grease, can fill crevices in a die, producing a weak strike with a smudged appearance. These errors are often called ""missing element coins"" (discussed above and as ""filled dies.""(A great example of such an error occurred in 1922, when only the Denver mint struck Lincoln cents. The dies were so covered in grease, because the mint had to speed up production, that the mintmark was obscured and therefore either nonexistent or weakened on the 1922 cents. These are very popular with collectors.) A uni-face coin results when two"
"Johnson circles In geometry, a set of Johnson circles comprises three circles of equal radius ""r"" sharing one common point of intersection ""H"". In such a configuration the circles usually have a total of four intersections (points where at least two of them meet): the common point ""H"" that they all share, and for each of the three pairs of circles one more intersection point (referred here as their 2-wise intersection). If any two of the circles happen to osculate, they only have ""H"" as a common point, and it will then be considered that ""H"" be their 2-wise intersection"
"an associative magic square, in which every pair of numbers symmetrically opposite to the center sum up to the same value, 26. For e.g., 16+10, 3+23, 6+20, etc. In the finished square, 1 is placed at center cell of bottom row, and successive numbers are placed via elongated knight's move (two cells right, two cells down). When a collision occurs, the break move is to move one cell up. Also note that all the odd numbers occur inside the central diamond formed by 1, 5, 25 and 21, while the even numbers are placed at the corners. The occurrence of"
"tips and 20 face centers. All 80 pieces can move relative to each other. There are also a good number of internal moving pieces inside the puzzle, which are necessary to keep it in one piece as its surface pieces are rearranged. There are two types of twists that it can undergo: a shallow twist which rotates the 5 triangles around a single tip, and a deep twist which rotates 5 entire faces (including the triangles around the tip) around the tip. The shallow twist moves the triangles between faces but keeps them around the same tip; the deeper twist"
"Circle packing In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that all circles touch one another. The associated packing density, ""η"", of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres. While the circle has a relatively low maximum packing density of 0.9069 on the Euclidean plane, it does not have the lowest possible. The ""worst"" shape"
"given here. There are 48,544 distinct 5 × 5 associative magic squares, up to rotation and reflection, such as the two below. There are no 6 × 6 (or 10 × 10) associative magic squares. Other than when there are none the precise number of associative magic squares at sizes bigger than 5 × 5 is unknown. The following table, due to Walter Trump, gives the sizes up to 10, followed by examples of 7 × 7 and 8 × 8 tables. Associative magic square An associative magic square is a magic square for which every pair of numbers symmetrically"
"and its third side – ""C"" ""C"" – has a side length of less than 2""r"". Therefore, the circles 1 and 2 intersect – a contradiction. In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. It is easy to arrange 12 spheres so that each touches a central sphere, but there is a lot of space left over, and it is not obvious that there is no way to pack in a 13th sphere. (In fact, there is so much extra space that any two"
"sphere, and has 20 pieces, all of them corners. The puzzle has twelve circles located at the vertices of the icosahedron, and on the six-color version opposite circles are the same color. Because of the rounded shape of the puzzle, the pieces move up and down as they are rotated. It is also possible to remove one piece, turning the puzzle into a spherical version of the 15 puzzle. The purpose of the puzzle is to scramble the colors, and then restore it to its original state of having one color per circle. This puzzle is equivalent to solving just"
"a pattern of four concentric rings appears on the table, with a small circle at its centre. Working inward, the zones are worth £500, £1,000, £2,000, £4,000, and £10,000. The player attempts to land one slider completely within each zone, in ascending order. If any portion of a slider lands within a zone other than the correct one, that slider is removed from play. Hitting the correct zone gives the player the option of either ending the game and keeping all winnings for that zone, or continuing the game and reusing the slider. Running out of sliders ends the round"
"4 corner pieces, which can be permutated in 4! different ways. They cannot be flipped; their orientation is fully determined by their location. The whole puzzle also has a parity constraint: the parity of the corner permutation is the same as the parity of the number of flipped edges. This divides the full number by 2. The full number of possible combinations on the Floppy Cube is therefore: In relative terms, this number is extremely low compared to other puzzles like the Rubik's Cube (with over 43 quintillion combinations), the Rubik's Domino (which has just under 410 million combinations), or"
"four dots placed in the form of a square at the centre of a circle on the other side; 2. the Horse on one side and a group of four dots placed in the form of a square at the centre of a circle on the other side; 3. the Bull on one side and a group of four dots placed in the form of a square at the centre of a circle on the other side. Also in the Northern and southern Sri Lanka, coins having a bull on one side and an elephant on the other side have"
"Coinage shapes Although the vast majority of coins are round, coins are made in a variety of other shapes, including, squares, diamonds, hexagons, heptagons, octagons, decagons, and dodecagons. They have also been struck with scalloped (wavy) edges, and with holes in the middle. Coins in the shape of polygons often have rounded edges or are Reuleaux polygons. This article focuses on circulating coins; a number of non-circulating commemorative coins have been made in special shapes, including guitars, pyramids, and maps. Indo-Greek coins were often square. Many countries have struck square coins with rounded corners. Some of these, such as the"
"cube assembled from eight unit cubes with magnets on their inside faces, allowing the cubes to rotate in groups of four around three axes. The object of the puzzle was to mix the colors on the faces of the cube and then restore them. After making many preliminary models, in 1968 a working prototype was constructed, and on April 11, 1972, U.S. patent 3,655,201 was issued covering the Nichols' Cube. The patent focused on the 2×2×2 puzzle but mentioned the possibility of larger versions. Nichols' patents became the subject of a court hearing, between his employer, and the Ideal Toy"
"The full number is 100 669 616 553 523 347 122 516 032 313 645 505 168 688 116 411 019 768 627 200 000 000 000 (roughly 101 unvigintillion on the short scale or 101 undecillion on the long scale). The corners are distinguishable on a 6-color Megaminx because two corners with the same three colors will be mirror images of each other. There are 15 pairs of identical edges. It would not be possible to swap all 15 pairs, since this would be an odd permutation of the edges, so a reducing factor of 2 is applied to"