Recreational mathematics

Recreational mathematics

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Queens problem

What is the maximum number of queens that can be placed on an chessboard such that no two attack one another? The answer is queens for or and queens otherwise, which gives eight queens for the usual board (Madachy 1979; Steinhaus 1999, p. 29). The number of different ways the queens can be placed on an chessboard so that no two queens may attack each other for the first few are 1, 0, 0, 2, 10, 4, 40, 92, ... (OEIS A000170; Madachy 1979; Steinhaus 1999, p. 29). The number of rotationally and reflectively distinct solutions of these are 1, 0, 0, 1, 2, 1, 6, 12, 46, 92, ... (OEIS A002562; Dudeney 1970; p. 96). The 12 distinct solutions for are illustrated above, and the remaining 80 are generated by rotation and reflection (Madachy 1979, Steinhaus 1999).The minimum number of queens needed to occupy or attack all squares of an chessboard (i.e., domination numbers for the queen graphs) are given for , 2, ... by 1, 1, 1, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 8, 9,..

Antelope graph

An antelope graph is a graph formed by all possible moves of a hypothetical chess piece called an "antelope" which moves analogously to a knight except that it is restricted to moves that change by three squares along one axis of the board and four squares along the other. To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable antelope moves are considered edges. The plots above show the graphs corresponding to antelope graph on chessboards for to 7.The antelope graph is connected for , Hamiltonian for (trivially) and 14 (but for no odd or other even values ), and traceable for and 21 (with the status for unknown and unknown).Precomputed properties of antelope graphs are implemented in the Wolfram Language as GraphData["Antelope", m, n]...

Giraffe graph

A giraffe graph is a graph formed by all possible moves of a hypothetical chess piece called a "giraffe" (a.k.a. -leaper) which moves analogously to a knight except that it is restricted to moves that change by one square along one axis of the board and four squares along the other. To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable giraffe moves are considered edges.The smallest board allowing a closed tour for the giraffe (i.e., the giraffe graph is Hamiltonian) is the , first solved by A. H. Frost in 1886.

Bishops problem

Find the maximum number of bishops that can be placed on an chessboard such that no two attack each other. The answer is (Dudeney 1970, Madachy 1979), giving the sequence 2, 4, 6, 8, ... (the even numbers) for , 3, .... One maximal solution for is illustrated above. The numbers of distinct maximal arrangements for , 2, ... bishops are 1, 4, 26, 260, 3368, ... (OEIS A002465). The numbers of rotationally and reflectively distinct solutions on an board for is(1)for (Dudeney 1970, p. 96; Madachy 1979, p. 45; Pickover 1995). An equivalent formula also valid for is(2)where is the floor function, giving the sequence for , 2, ... as 1, 1, 2, 3, 6, 10, 20, 36, ... (OEIS A005418).The minimum number of bishops needed to occupy or attack all squares on an chessboard is , arranged as illustrated above...

Magic tesseract

A magic tesseract is a four-dimensional generalization of the two-dimensional magic square and the three-dimensional magic cube. A magic tesseract has magic constantso for , 2, ..., the magic tesseract constants are 1, 17, 123, 514, 1565, 3891, ... (OEIS A021003).Berlekamp et al. (1982, p. 783) give a magic tesseract. J. Hendricks has constructed magic tesseracts of orders three, four, five (Hendricks 1999a, pp. 128-129), and six (Heinz). M. Houlton has used Hendricks' techniques to construct magic tesseracts of orders 5, 7, and 9.There are 58 distinct magic tesseracts of order three, modulo rotations and reflections (Heinz, Hendricks 1999), one of which is illustrated above. Each of the 27 rows (e.g., 1-72-50), columns (e.g., 1-80-42), pillars (e.g., 1-54-68), and files (e.g., 1-78-44) sum to the magic constant 123.Hendricks (1968) has constructed a pan-4-agonal magic tesseract of order 4. No pan-4-agonal..

Nested polygon

Beautiful patterns can be created by drawing sets of nested polygons such that the incircle of the th polygon is the circumcircle of the st and successive polygons are rotated one half-turn at each iteration. The results are shown above for nested triangles through heptagons in alternating black and white and in a cyclic rainbow coloring.The animation above shows successive iterations of a nested octagon.The black region of a nested square has areaif the initial square has unit edge length.

Bill picture

A Bill picture is a sequence of nested regular polygons in which subsequent polygons are each rotated so that they begin one vertex further. The term was coined by Trott (2004, pp. 88-89) and commemorates Swiss artist Max Bill, who in 1938 created a picture showing a similar arrangement of the equilateral triangle through octagon (Huttingerr 1978, Bill 1987).The figure above shows the Bill picture including regular polygons up through theregular dodecagon.

Truchet tiling

In 1704, Sebastien Truchet considered all possible patterns formed by tilings of right triangles oriented at the four corners of a square (Wolfram 2002, p. 875).Truchet's tiles produce beautiful patterns when laid out on a grid, as illustrated by the arrangement of random tiles illustrated above.A modification of Truchet's tiles leads to a single tile consisting of two circular arcs of radius equal to half the tile edge length centered at opposed corners (Pickover 1989). There are two possible orientations of this tile, and tiling the plane using tiles with random orientations gives visually interesting patterns. In fact, these tiles have been used in the construction of various games, including the "black path game" and "meander" (Berlekamp et al. 1982, pp. 682-684).The illustration above shows a Truchet tiling. For random orientations, the fraction of closed circles is approximately 0.054 and the..

Magic square

A magic square is a square array of numbers consisting of the distinct positive integers 1, 2, ..., arranged such that the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same number (Kraitchik 1942, p. 142; Andrews 1960, p. 1; Gardner 1961, p. 130; Madachy 1979, p. 84; Benson and Jacoby 1981, p. 3; Ball and Coxeter 1987, p. 193), known as the magic constantIf every number in a magic square is subtracted from , another magic square is obtained called the complementary magic square. A square consisting of consecutive numbers starting with 1 is sometimes known as a "normal" magic square.The unique normal square of order three was known to the ancient Chinese, who called it the Lo Shu. A version of the order-4 magic square with the numbers 15 and 14 in adjacent middle columns in the bottom row is called Dürer's magic square. Magic squares of order 3 through 8 are shown..

Perfect magic cube

A perfect magic cube is a magic cube for which the rows, columns, pillars, space diagonals, and diagonals of each orthogonal slice sum to the same number (i.e., the magic constant ). While this terminology is standard in the published literature (Gardner 1976, Benson and Jacoby 1981, Gardner 1988, Pickover 2002), it has been suggested at various times that such cubes instead be termed Myers cubes, Myers diagonal cubes, or diagonal magic cube (Heinz).There is a trivial perfect magic cube of order one, but no perfect cubes exist for orders 2-4 (Schroeppel 1972; Benson and Jacoby 1981, pp. 23-25; Gardner 1988). While normal perfect magic cubes of orders 7 and 9 have been known since the late 1800s, it was long not known if perfect magic cubes of orders 5 or 6 could exist (Wells 1986, p. 72), although Schroeppel (1972) and Gardner (1988) note that any such cube must have a central value of 63. (Confusingly, Benson and Jacoby (1981, p. 5)..


Checkers is a two-player game with the most common variant played on an checkerboard with each player starts with twelve pieces of a fixed color on opposite sites of the board. The most common variant of checkers is so-called "pool checkers," also called "Spanish pool checkers," draughts or draught (in the United Kingdom and some other countries), American checkers, and straight checkers. Play proceeds alternately between players, where all pieces may initially only move and capture in a forward diagonal direction. The allowable direction of play is modified for a piece if it is "crowned" by reaching the other side of the board, after which it may move either forwards or backwards. An opponent's piece may be captured by jumping over it diagonally, and the game is won by capturing all the opponents pieces or leaving the opponent with no legal moves.The most widely available sets of checkers consist of black and..

Kings problem

The problem of determining how many nonattacking kings can be placed on an chessboard. For , the solution is 16, as illustrated above (Madachy 1979). In general, the solutions are(1)(Madachy 1979), giving the sequence of doubled squares 1, 1, 4, 4, 9, 9, 16, 16, ... (OEIS A008794). This sequence has generating function(2)The minimal number of kings needed to occupy or attack every square on an chessboard (i.e., domination numbers for the king graphs) are given for , 2, ... by 1, 1, 1, 4, 4, 4, 9, 9, 9, 16, ... (OEIS A075561), with the case illustrated above and noted by (Madachy 1979, p. 39). In general, for an chessboard,(3)

Möbius strip dissection

Tiling of a Möbius strip can be performed immediately by carrying over a tiling of a rectangle with the same two-sided surface area. However, additional tilings are possible by cutting tiles across glued edges. An example of such a tiling is the strip constructed from a rectangle consisting of two halves of a width 2 square (which are rejoined when edges are connected) separated by a square (Stewart 1997). Unfortunately, since the long top and bottom edges must be glued together, this example is not constructible out of paper. It also suffers from having the unit square share a boundary with itself. In 1993, S. J. Chapman found a tiling free of the latter defect (although still suffering from the former) which can be constructed using five squares. No similar tiling is possible using fewer tiles (Stewart 1997)...

Curry triangle

The Curry triangle, also sometimes called the missing square puzzle, is a dissection fallacy created by American neuropsychiatrist L. Vosburgh Lions as an example of a phenomenon discovered by Paul Curry. The figure apparently shows that a triangle of area 60, a triangle of area 58 containing a rectangular hole, and a broken rectangle of area 59 can all be formed out of the same set of 6 polygonal pieces. The explanation for this lies in the inaccuracy of the initial subdivision. In the diagrams, the small and large right triangles are similar, hence they cannot have perpendicular sides of lengths and , respectively, as apparently shown in the drawing.


The stomachion is a 14-piece dissection puzzle similar to tangrams. It is described in fragmentary manuscripts attributed to Archimedes as noted by Magnus Ausonius (310-395 A.D.). The puzzle is also referred to as the "loculus of Archimedes" (Archimedes' box) or "syntemachion" in Latin texts. The word stomachion has as its root the Greek word , meaning "stomach." Note that Ausonius refers to the figure as the "ostomachion," an apparent corruption of the original Greek.The puzzle consists of 14 flat pieces of various shapes arranged in the shape of a square, with the vertices of pieces occurring on a grid. Two pairs of pieces are duplicated. Like tangrams, the object is to rearrange the pieces to form interesting shapes such as the elephant illustrated above (Andrea).Taking the square as having edge lengths 12, the pieces have areas 3, 3, 6, 6, 6, 6, 9, 12, 12, 12, 12, 12, 21, and 24, giving them relative..

Plato's numbers

The positive integers 216 and appear in an obscure passage in Plato's The Republic. In this passage, Plato alludes to the fact that 216 is equal to , where 6 is one of the numbers representing marriage since it is the product to the female 2 and the male 3. Plato was also aware of the fact the sum of the cubes of the 3-4-5 Pythagorean triple is equal to (Livio 2002, p. 66).In Laws, Plato suggests that is the optimal number of citizens in a state because 1. It is the product of 12, 20, and 21. 2. The 12th part of it can still be divided by 12. 3. It has 59 proper divisors, including all numbers for 1 to 12 except 11, and 5038--which is very close to 5040--is divisible by 11 (Livio 2002, p. 65).

Möbius net

The perspective image of an infinite checkerboard. It can be constructed starting from any triangle , where and form the near corner of the floor, and is the horizon (left figure). If is the corner tile, the lines and must be parallel to and respectively. This means that in the drawing they will meet and at the horizon, i.e., at point and point respectively (right figure). This property, of course, extends to the two bunches of perpendicular lines forming the grid.The adjacent tile (left figure) can then be determined by the following conditions: 1. The new vertices and lie on lines and respectively. 2. The diagonal meets the parallel line at the horizon . 3. The line passes through . Similarly, the corner-neighbor of (right figure) can be easily constructed requiring that: 1. Point lie on . 2. Point lie on the common diagonal of the two tiles. 3. Line pass through . Iterating the above procedures will yield the complete picture. This construction shows..

Exchange shuffle

A shuffle of a deck of cards obtained by successively exchanging the cards in position 1, 2, ..., with cards in randomly chosen positions. For , the most frequent permutation is , where if is even and either or if is odd (Goldstine and Moews 2000). Amazingly, for cards, the identity permutation (i.e., the original state before the cards were shuffled) is the most likely (Goldstein and Moews 2000).

Soma cube

A solid dissection puzzle invented by Piet Hein during a lecture on Quantum Mechanics by Werner Heisenberg. There are seven soma pieces composed of all the irregular face-joined cubes (polycubes) with cubes. The object is to assemble the pieces into a cube. There are 240 essentially distinct ways of doing so (Beeler 1972, Berlekamp et al. 1982), as first enumerated one rainy afternoon in 1961 by J. H. Conway and Mike Guy.A commercial version of the cube colors the pieces black, green, orange, white, red, and blue. When the 48 symmetries of the cube, three ways of assembling the black piece, and ways of assembling the green, orange, white, red, and blue pieces are counted, the total number of solutions rises to .

Haberdasher's problem

With three cuts, dissect an equilateral triangle into a square. The problem was first proposed by Dudeney in 1902, and subsequently discussed in Dudeney (1958), and Gardner (1961, p. 34), Stewart (1987, p. 169), and Wells (1991, pp. 61-62). The solution can be hinged so that the four pieces collapse into either the triangle or the square. Two of the hinges bisect sides of the triangle, while the third hinge and the corner of the large piece on the base cut the base in the approximate ratio 0.982:2:1.018.


An official chessboard is an board containing squares alternating in color between olive green and buff (where "buff" is a color variously defined as a moderate orange yellow or a light to moderate yellow) on which the game of chess is played. The checkerboard is identical to the chessboard, and in both cases, the squares are referred to as "black" and "white." In chess (as in checkers), the board is oriented so that each player has a black square on his left.It is impossible to cover a chessboard from which two opposite corners have beenremoved with dominoes.The above plot shows a chessboard centered at (0, 0) and its inverse about a small circle also centered at (0, 0) (Gardner 1984, pp. 244-245; Dixon 1991).The illustration above shows an infinite chessboard reflected in a sphere...

Franklin magic square

In 1750, Benjamin Franklin constructed the above semimagic square having magic constant 260. Any half-row or half-column in this square totals 130, and the four corners plus the middle total 260. In addition, bent diagonals (such as 52-3-5-54-10-57-63-16) also total 260 (Madachy 1979, p. 87).Describing his invention in 1771, Franklin stated, "I was at length tired with sitting there to hear debates, in which, as clerk, I could take no part, and which were often so unentertaining that I was induc'd to amuse myself with making magic squares or circles" (Franklin 1793).

Dürer's magic square

Dürer's magic square is a magic square with magic constant 34 used in an engraving entitled Melancholia I by Albrecht Dürer (The British Museum, Burton 1989, Gellert et al. 1989). The engraving shows a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Dürer's magic square is located in the upper right-hand corner of the engraving. The numbers 15 and 14 appear in the middle of the bottom row, indicating the date of the engraving, 1514.Dürer's magic square has the additional property that the sums in any of the four quadrants, as well as the sum of the middle four numbers, are all 34 (Hunter and Madachy 1975, p. 24). It is thus a gnomon magic square. In addition, any pair of numbers symmetrically placed about the center of the square sums to 17, a property making the square even more magical...

Trimagic square

If replacing each number by its square or cube in a magic square produces another magic square, the square is said to be a trimagic square. Trimagic squares are also called trebly magic squares, and are 3-multimagic squares.Trimagic squares of order 12, 32, and larger are known. Tarry (1906) gave a method for constructing a trimagic square of order 128, Cazalas a method for trimagic squares of orders 64 and 81, R. V. Heath a method for constructing an order 64 trimagic square which is different from Cazalas's (Kraitchik 1942), and Benson (Benson and Jacoby 1976) a method for constructing an order 32 trimagic square.Walter Trump constructed the first trimagic square of order 12 in June 2002. This square, illustrated above, is the smallest possible trimagic square, since Boyer and Trump subsequently proved that a trimagic square of order less than 12 cannot exist (Boyer)...

Multiplication magic square

A square which is magic under multiplication instead of addition (the operation used to define a conventional magic square) is called a multiplication magic square. Unlike (normal) magic squares, the entries for an th order multiplicative magic square are not required to be consecutive.The above multiplication magic square has a multiplicative magic constant of 4096 and was found by Antoine Arnauld in Nouveaux Eléments de Géométrie, Paris in 1667 (Boyer).The smallest possible magic constants for , , ... are 216, 5040, 302400, 25945920, ... (OEIS A114060). The solution (left) was found by Sayles in 1913 and also published by Dudeney (1917). Sayles also found the solution (right), which was subsequently proved to be minimal by Borkovitz and Hwang (1983). The series of best known smallest largest element for an multiplication magic square with , 4, ... begins 36, 28, 45, 66, 91, 160, 225, ... (Boyer)...

Multimagic square

A magic square is said to be -multimagic if the square formed by replacing each element by its th power for , 2, ..., is also magic. A 2-multimagic square is called bimagic, a 3-multimagic square is called trimagic, a 4-multimagic square is called tetramagic, a 5-multimagic square is called pentamagic, and so on.The first known bimagic square had order eight and was constructed by Pfefferman (1891). Tetramagic and pentamagic squares were constructed by Christian Boyer and André Viricel in 2001 (Boyer 2001).

Border square

A magic square that remains magic when its border is removed. A nested magic square remains magic after the border is successively removed one ring at a time. An example of a nested magic square is the order 7 square illustrated above (i.e., the order 7, 5, and 3 squares obtained from it are all magic).The amazing square above (Madachy 1979, pp. 93-94) is a prime magic border square, so that the , , ..., and subsquares are all also prime magic squares.

Semimagic square

A semimagic square is a square that fails to be a magic square only because one or both of the main diagonal sums do not equal the magic constant (Kraitchik 1942, p. 143). Note some care with terminology is necessary. For example, Jelliss terms a semimagic square a "magic square" and a magic square a "diagonally magic square."The number of distinct semimagic squares (treating squares differing by rotations and reflections as identical) of orders 1, 2, ... are 1, 0, 8, .... The eight semimagic squares of order 3 are illustrated above.

Associative magic square

An magic square for which every pair of numbers symmetrically opposite the center sum to . The Lo Shu is associative but not panmagic. The numbers of associative magic squares of order 1, 2, ... are 1, 0, 1, 48, ... (OEIS A081262).Order four squares can be panmagic or associative, but not both. Order five squares are the smallest which can be both associative and panmagic, and 16 distinct associative panmagic squares exist, one of which is illustrated above (Gardner 1988). The numbers of associative panmagic squares of order 1, 2, ... are therefore 1, 0, 0, 0, 16, ... (OEIS A081263).

Panmagic square

If all the diagonals--including those obtained by "wrapping around" the edges--of a magic square sum to the same magic constant, the square is said to be a panmagic square (Kraitchik 1942, pp. 143 and 189-191). (Only the rows, columns, and main diagonals must sum to the same constant for the usual type of magic square.) The terms diabolic square (Gardner 1961, pp. 135-137; Hunter and Madachy 1975, p. 24; Madachy 1979, p. 87), pandiagonal square (Hunter and Madachy 1975, p. 24), and Nasik square (Madachy 1979, p. 87) are sometimes also used.No panmagic squares exist of order 3 or any order for an integer. The Siamese method for generating magic squares produces panmagic squares for orders with ordinary vector (2, 1) and break vector (1, ).The Lo Shu is not panmagic, but it is an associative magic square. Order four squares can be panmagic or associative, but not both. Order five squares are the smallest..

Alphamagic square

A magic square for which the number of letters in the word for each number generates another magic square. This definition depends, of course, on the language being used. In English, for example,where the magic square on the right corresponds tothe number of letters in

Gnomon magic square

A magic square in which the elements in each corner have the same sum. Dürer's magic square, illustrated above, is an example of a gnomon magic square since the sums in any of the four quadrants (as well as the sum of the middle four numbers) are all 34 (Hunter and Madachy 1975, p. 24).

Multimagic series

A set distinct numbers taken from the interval form a magic series if their sum is the th magic constant(Kraitchik 1942, p. 143). If the sum of the th powers of these numbers is the magic constant of degree for all , then they are said to form a th order multimagic series. Here, the magic constant of degree is defined as times the sum of the first th powers,where is a generalized harmonic number of order .For example is bimagic since and . It is also trimagic since . Similarly, is trimagic.The numbers of magic series of various lengths are gives in the following table for small orders (Kraitchik 1942, p. 76; Boyer), where the , 3, and 4 values were corrected and extended by Boyer and Trump in 2002.SloaneA052456A052457A052458A090037111112200038000486220513948206321349800795733218440083515434038039121091537408202949738126010781325415282464323600114528684996756947689757311870122950111860062822226896106..

Magic series

A set of distinct numbers taken from the interval form a magic series if their sum is the th magic constant(Kraitchik 1942, p. 143). The numbers of magic series of orders , 2, ..., are 1, 2, 8, 86, 1394, ... (OEIS A052456). The following table gives the first few magic series of small order.magic series12, 3, , , , , , , If the sum of the th powers of these number is the magic constant of degree for all , then they are said to form a th order multimagic series. Here, the magic constant of degree is defined as times the sum of the first th powers,where is a harmonic number of order .

Tetramagic cube

A tetramagic cube is a magic cube that remains magicwhen all its numbers are squared, cubed, and taken to the fourth power.Only two tetramagic cubes are known, and both were found by C. Boyer in 2003. The smaller of these is a tetramagic cube of order 1024; this cube, its square, and its cube are perfect, while its fourth power is only semiperfect. The larger is a perfect tetramagic cube of order 8192; this cube, its square, its cube, and its fourth power are perfect.

Semiperfect magic cube

A semiperfect magic cube, sometimes also called an "Andrews cube" (Gardner 1976; Gardner 1988, p. 219) is a magic cube for which the cross section diagonals do not sum to the magic constant. Some care is needed with terminology, as some authors drop the "semiperfect" and refer to such cubes simple as "magic cubes" (e.g., Benson and Jacoby 1981, p. 4).A semiperfect magic cube of order 3 has magic constant 42. It must be associative with opposite elements summing to (Andrews 1960, p. 65) and have as its center (Gardner 1976; Benson and Jacoby 1981, p. 4; Andrews 1960, p. 65). Hendricks (1972) proved that there are four distinct semiperfect magic cubes excluding rotations and reflections (Gardner 1976; Benson and Jacoby 1981, pp. 4 and 11-13), illustrated above. These cubes were described by Andrews (1960, pp. 66-70), although he seems not to have noted that they represent..

Pandiagonal semiperfect magic cube

A pandiagonal semiperfect magic cube is a semiperfect magic cube that remains semiperfect when any single orthogonal section is "restacked" cyclically so that the ordering of any set of plane sections becomes , , , ..., or (Benson and Jacoby 1981, p. 4).There is no pandiagonal semiperfect magic cube of order 3 (Benson and Jacoby 1988, p. 14).The order 4 and 5 magic cubes shown above are pandiagonal semiperfect magic (Benson and Jacoby 1988, pp. 15-23 and 30).

Pandiagonal perfect magic cube

A pandiagonal perfect magic cube is a perfect magic cube that remains perfect when any single orthogonal section is "restacked" cyclically so that the ordering of any set of plane sections becomes , , , ..., or (Benson and Jacoby 1981, p. 4).Pandiagonal perfect magic cubes are possible for orders 8 and 9, but no smaller orders. They are also not possible for orders 12 or 14, but are possible for all orders that are multiples of 8 and odd orders greater than or equal to 9 (Benson and Jacoby 1981, p. 5). Benson and Jacoby (1981, pp. 76-78) explicitly construct a pandiagonal perfect magic cube of order 9.Planck (1950; cited in Gardner 1988) constructed a perfect pandiagonal magic cube.

Multimagic cube

An -fold multimagic cube is a magic cube that remains magic when each element is squared, cubed, etc., up to th power. (Of course, when the elements of a cube are taken to a power, the numbers are no longer consecutive, so the resulting magic cube is no longer simple.) A 2-fold multimagic cube is called a bimagic cube, a 3-fold multimagic cube is called a trimagic cube, a 4-fold multimagic cube is called a tetramagic cube, and so on.

Magic cube

A magic cube is an version of a magic square in which the rows, columns, pillars, and four space diagonals each sum to a single number known as the cube's magic constant. Magic cubes are most commonly assumed to be "normal," i.e., to have elements that are the consecutive integers 1, 2, ..., . However, this requirement is dropped (as it must be) in the consideration of so-called multimagic cubes.If it exists, a normal magic cube has magic constantFor , 2, ..., the magic constants are given by 1, 9, 42, 130, 315, 651, ... (OEIS A027441).If only rows, columns, pillars, and space diagonals sum to , a magic cube is called a semiperfect magic cube, or sometimes an Andrews cube (Gardner 1988, p. 219). If, in addition, the diagonals of each orthogonal slice sum to , then the magic cube is called a perfect magic cube. If a perfect or semiperfect magic cube is magic not only along the main space diagonals, but also on the broken space diagonals, it is..

Bimagic cube

A bimagic cube is a (normal) magic cube that remains magic when all its elements are squared. Of course, even a normal magic cubic becomes nonnormal (i.e., contains nonconsecutive elements) upon squaring.Cazalas (1934) attempted but failed to construct a bimagic cube (Boyer). David M. Collison apparently constructed a bimagic cube of order 25 in an unpublished paper (Hendricks 1992), but it was not until the year 2000 that John Hendricks published an order 25 perfect magic cube whose square is a semiperfect magic cube.On January 20, 2003, Christian Boyer discovered an order 16 bimagic cube (where the cube itself is perfect magic, but its square is only semiperfect magic). This was rapidly followed by another order 16 bimagic cube (where the base cube is perfect and its square semiperfect) on January 23, an order 32 bimagic cube (where both the base cube and its square are perfect) on January 27, and an order 27 bimagic cube (where the base..

Magic constant

The number(1)(2)to which the numbers in any horizontal, vertical, or main diagonal line must sum in a magic square. The first few values are 1, 5, 15, 34, 65, 111, 175, 260, ... (OEIS A006003). The magic constant for an th order magic square starting with an integer and with entries in an increasing arithmetic series with difference between terms is(3)(Hunter and Madachy 1975, Madachy 1979). In a panmagic square, in addition to the main diagonals, the broken diagonals also sum to .For a magic cube, magic tesseract, etc., the magic -D constant is(4)(5)The first few magic constants are summarized in the following table.SloaneA006003A027441A021003111125917315421234341305145653151565There is a corresponding multiplicative magic constant for multiplicationmagic squares.A similar magic constant of degree is defined for magic series and multimagic series as times the sum of the first th powers,(6)(7)where is a harmonic number of order..

String figure

A string figure is any pattern produced when a looped string is spanned between two hands and is twisted and woven in various manners around the fingers and the wrists. The combinations of crossings which can be realized in this way can be studied using knot theory.The string figure above is known as the Apache door (Jayne 1975, pp. 12-15, Fig. 21) or tent flap (Ball 1971, p. 5, Fig. 2).The string figure illustrated above is known as "Jacob's ladder," Osage diamonds (Jayne 1975, pp. 24-27, Fig. 50), the fishing net, or quadruple diamonds (Ball 1971, p. 19, Fig. 7).String figures, which belong to the ancient traditions of many peoples around the world, and are even present in primitive cultures, are nowadays considered as a recreational activity in mathematics education. In English-speaking countries they are also known as the children's game called "cat's cradle."..


A maze, also known as a labyrinth, as is a set of passages (with impermeable walls). The goal of the maze is to start at one given point and find a path through the passages that reaches a second given point.The back of a clay accounting tablet from Pylos, Greece is illustrated above (Wolfram 2002, p. 43). Legend has it that it was the plan for the labyrinth housing the minotaur in the palace at Knossos, Crete, and that it was designed by Daedalus. It is also said that it was a logo for the city of Troy-or perhaps the plan of some of its walls (Wolfram 2002, p. 873).The above pattern (in either its square or rounded form) has appeared with remarkably little variation in a large variety of places all over the world-from Cretan coins, to graffiti at Pompeii, to the floor of the cathedral at Chartres, to carvings in Peru, to logos for aboriginal tribes. For probably three thousand years, it has been the single most common design used for mazes (Wolfram..

Icosian game

The Icosian game, also called the Hamiltonian game (Ball and Coxeter 1987, p. 262), is the problem of finding a Hamiltonian cycle along the edges of an dodecahedron, i.e., a path such that every vertex is visited a single time, no edge is visited twice, and the ending point is the same as the starting point (left figure). The puzzle was distributed commercially as a pegboard with holes at the nodes of the dodecahedral graph, illustrated above (right figure). The Icosian Game was invented in 1857 by William Rowan Hamilton. Hamilton sold it to a London game dealer in 1859 for 25 pounds, and the game was subsequently marketed in Europe in a number of forms (Gardner 1957).A graph having a Hamiltonian cycle, i.e., on which the Icosian game may be played, is said to be a Hamiltonian graph. While the skeletons of all the Platonic solids and Archimedean solids (i.e., the Platonic graphs and Archimedean graphs, respectively) are Hamiltonian, the same is..


Yahtzee is a game played with five 6-sided dice. Players take turns rolling the dice, and trying to get certain types of rolls, each with an assigned point value, as summarized in the following table. Players are allowed a total of three rolls, with any subset of dice capable of being set aside at each roll. In addition to runs of a single number, other rolls include 3 of a kind (three of the same number), 4 of a kind (four of the same number), full house (two of one number and three of another), small straight (4 numbers in a row), large straight (5 numbers in a row), Yahtzee (five of the same number), and chance (any roll).acessum of 1stwossum of 2sthreessum of 3sfourssum of 4sfivessum of 5ssixessum of 6s3 of a kindsum of all dice4 of a kindsum of all dicefull house25sm. straight30lg. straight40Yahtzee50chancesum of all diceIn addition to points scored by obtaining various rolls of the dice, 35 bonus points are awarded if the total score from the first six categories..

Sicherman dice

A pair of dice which have the same odds for throwing every number as a normal pair of 6-sided dice. They are the only such alternate arrangement if face values are required to be positive. However, if faces are permitted to have zero value (i.e., to be blank), then two additional possible equal-odds pairs of dice are obtained by subtracting one from each face on either of the two dice and adding one to each face the other. If negative values are permitted, there are an infinite number of equal-odds dice.

Efron's dice

Efron's dice are set of four nontransitive dice such that the probabilities of A winning against B, B against C, C against D, and D against A are all the same.The images above depict two different sets of Efron's dice having 2:1 odds for winning pairs.Another set of dice in which ties may occur (in which case the dice are rolled again) and which gives odds of 11:6 for winning pairs is illustrated above.


A die (plural "dice") is a solid with markings on each of its faces. The faces are usually all the same shape, making Platonic solids and Archimedean duals the obvious choices. The die can be "rolled" by throwing it in the air and allowing it to come to rest on one of its faces. Dice are used in many games of chance as a way of picking random numbers on which to bet, and are used in board or role-playing games to determine the number of spaces to move, results of a conflict, etc. A coin can be viewed as a degenerate 2-sided case of a die.In 1787, Mozart wrote the measures and instructions for a musical composition dice game. The idea is to cut and paste pre-written measures of music together to create a Minuet (Chuang).The most common type of die is a six-sided cube with the numbers 1-6 placed on the faces. The value of the roll is indicated by the number of "spots" showing on the top. For the six-sided die, opposite faces are arranged..


A game played with two dice. If the total is 7 or 11 (a "natural"), the thrower wins and retains the dice for another throw. If the total is 2, 3, or 12 ("craps"), the thrower loses but retains the dice. If the total is any other number (called the thrower's "point"), the thrower must continue throwing and roll the "point" value again before throwing a 7. If he succeeds, he wins and retains the dice, but if a 7 appears first, the player loses and passes the dice.The following table summarizes the probabilities of winning on a roll-by-roll basis, where is the probability of rolling a point . For rolls that are not naturals (W) or craps (L), the probability that the point will be rolled first is found from(1)(2)W/L2L03L04567W1891011W112L0Summing from to 12 then gives the probability of winning as (Kraitchik 1942; Mosteller 1987, p. 26), just under 50%. Two and 12 are the hardest sums to roll, since each can..


A roll of two 6s (the highest roll possible) on a pair of 6-sided dice.The probability of rolling boxcars in a single roll of two dice is 1/36, or 2.777...%.In order to have a 50% chance of obtaining at least one boxcars in rolls of two dice, it must be true that(1)so solving for gives(2)In fact, rolling two dice 25 times gives a probability of(3)that at least once boxcars will occur.

Riffle shuffle

A riffle shuffle, also called the Faro shuffle, is a shuffle in which a deck of cards is divided into two halves. The top half of the deck is placed in the left hand, and cards are then alternatively interleaved from the left and right hands (an in-shuffle) or from the right and left hands (an out-shuffle). Using an in-shuffle, a deck originally arranged as 1 2 3 4 5 6 7 8 would become 5 1 6 2 7 3 8 4. Using an out-shuffle, the deck order would become 1 5 2 6 3 7 4 8. Riffle shuffles are used in card tricks (Marlo 1958ab, Adler 1973), and also in the theory of parallel processing (Stone 1971, Chen et al. 1981).The riffle operation is implemented in the WolframLanguage as: Riffle @@ Partition[Range[n], n/2, n/2, 1, {}]In general, card moves to the position originally occupied by the th card modulo . For an in-shuffle, the first card is numbered 1 and the multiplication is done modulo . For an out-shuffle, the first card is numbered 0 and the multiplication is done modulo..


Poker is a card game played with a normal deck of 52 cards. Sometimes, additional cards called "jokers" are also used. In straight or draw poker, each player is normally dealt a hand of five cards. Depending on the variant, players then discard and redraw cards, trying to improve their hands. Bets are placed at each discard step. The number of possible distinct five-card hands is equal to the number of possible ways of picking five cards out of a deck of 52, namelywhere is a binomial coefficient.There are special names for specific types of hands. A royal flush is an ace, king, queen, jack, and 10, all of one suit. A straight flush is five consecutive cards all of the same suit (but not a royal flush), where an ace may count as either high or low. A full house is three-of-a-kind and a pair. A flush is five cards of the same suit (but not a royal flush or straight flush). A straight is five consecutive cards (but not a royal flush or straight flush), where..

Monge's shuffle

A shuffle in which cards from the top of the deck in the left hand are alternatively moved to the bottom and top of the deck in the right hand. If the deck is shuffled times, the final position and initial position of a card are related byfor a deck of cards (Kraitchik 1942).


Cards are a set of rectangular pieces of cardboard with markings on one side and a uniform pattern on the other. The collection of all cards is called a "deck," and a normal deck of cards consists of 52 cards having 13 distinct values for each of four different "suits." The suits are called clubs (), diamonds (), hearts (), and spades (). Spades and clubs are colored black, while hearts and diamonds are colored red. The cards of each suit are numbered 1 through 13, where the special terms ace (1), jack (11), queen (12), and king (13) are used instead of numbers 1 and 11-13. However, in bridge and a number of other games, the ace is considered the highest card, and so would be assigned a value of 14 instead of 1.The randomization of the order of cards in a deck is called shuffling. Cards are used in many gambling games (such as poker), and the investigation of the probabilities of various outcomes in card games was one of the original motivations..


Bridge is a card game played with a normal deck of 52 cards.The number of possible distinct 13-card hands is(1)where is a binomial coefficient. While the chances of being dealt a hand of 13 cards (out of 52) of the same suit are(2)(Mosteller 1987, p. 8), the chance that at least one of four players will receive a hand of a single suit is(3)while the probability that exactly one person will have a perfect hand is(4)(Gridgeman 1964; Mosteller 1987, p. 8).There are special names for specific types of hands. A ten, jack, queen, king, or ace is called an "honor." Getting the three top cards (ace, king, and queen) of three suits and the ace, king, and queen, and jack of the remaining suit is called 13 top honors. Getting all cards of the same suit is called a 13-card suit. Getting 12 cards of same suit with ace high and the 13th card not an ace is called 2-card suit, ace high. Getting no honors is called a Yarborough.The probabilities of being..

Kawasaki's theorem

A theorem giving a criterion for an origami construction to be flat. Kawasaki's theorem states that a given crease pattern can be folded to a flat origami iff all the sequences of angles , ..., surrounding each (interior) vertex fulfil the following conditionNote that the number of angles is always even; each of them corresponds to a layer of the folded sheet.The rule evidently applies to the case of a rectangular sheet of paper folded twice, where the crease pattern is formed by the bisectors. But there are many more interesting examples where the above property can be checked (see, for example, the crane origami in the above figure).

Stamp folding

The number of ways of folding a strip of stamps has several possible variants. Considering only positions of the hinges for unlabeled stamps without regard to orientation of the stamps, the number of foldings is denoted . If the stamps are labeled and orientation is taken into account, the number of foldings is denoted . Finally, the number of symmetric foldings is denoted . The following table summarizes these values for the first .SloaneA001010A001011A0001361111221232264451656145068381447181204628203531392956114845361048352714060

Map folding

A general formula giving the number of distinct ways of folding an rectangular map is not known. A distinct folding is defined as a permutation of numbered cells reading from the top down. Lunnon (1971) gives values up to .OEIS, , ...1A0001361, 2, 6, 16, 50, 144, 462, 1392, ...2A0014152, 8, 60, 320, 1980, 10512, ...The number of ways to fold an sheet of maps is given for , 2, ..., etc. by 1, 8, 1368, 300608, 186086600, ... (Lunnon 1971; OEIS A001418).The limiting ratio of the number of strips to the number of strips is given by


A flexagon-like structure created by connecting the ends of a strip of four squares after folding along diagonals. Using a number of folding movements, it is possible to flip the flexatube inside out so that the faces originally facing inward face outward. Gardner (1961) illustrated one possible solution, and Steinhaus (1999) gives a second.

Wheat and chessboard problem

Let one grain of wheat be placed on the first square of a chessboard, two on the second, four on the third, eight on the fourth, etc. How many grains total are placed on an chessboard? Since this is a geometric series, the answer for squares isa Mersenne number. Plugging in then gives .

Bimagic square

If replacing each number by its square in a magic square produces another magic square, the square is said to be a bimagic square. Bimagic squares are also called doubly magic squares, and are 2-multimagic squares.Lucas (1891) and later Hendricks (1998) showed that a bimagic square of order 3 is impossible for any set of numbers except the trivial case of using the same number 9 times.The first known bimagic square, constructed by Pfeffermann (1891a; left figure), had order 8 with magic constant 260 for the base square and after squaring. Another order 8 bimagic square is shown at right.Benson and Jacoby (1976) stated their belief that no bimagic squares of order less than 8 exist, and this was subsequently proved by Boyer and Trump in 2002 (Boyer).Pfeffermann (1891b) also published the first 9th-order bimagic square. Only a part of the first Pfeffermann's bimagic squares of both order 8 and of order 9 were published, with their completion left as..


An unfolding is the cutting along edges and flattening out of a polyhedron to form a net. Determining how to unfold a polyhedron into a net is tricky. For example, cuts cannot be made along all edges that surround a face or the face will completely separate. Furthermore, for a polyhedron with no coplanar faces, at least one edge cut must be made from each vertex or else the polyhedron will not flatten. In fact, the edges that must be cut corresponds to a special kind of graph called a spanning tree of the skeleton of the polyhedron (Malkevitch).In 1987, K. Fukuda conjectured that no convex polyhedra admit a self-overlapping unfolding. The top figure above shows a counterexample to the conjecture found by M. Namiki. An unfoldable tetrahedron was also subsequently found (bottom figure above). Another nonregular convex polyhedra admitting an overlapping unfolding was found by G. Valette (shown in Buekenhout and Parker 1998).Examples..

Prime magic square

A prime magic square is a magic square consisting only of prime numbers (although the number 1 is sometimes allowed in such squares). The left square is the prime magic square (containing a 1) having the smallest possible magic constant, and was discovered by Dudeney in 1917 (Dudeney 1970; Gardner 1984, p. 86). The second square is the magic square consisting of primes only having the smallest possible magic constant (Madachy 1979, p. 95; attributed to R. Ondrejka). The third square is the prime magic square consisting of primes in arithmetic progression () having the smallest possible magic constant of 3117 (Madachy 1979, p. 95; attributed to R. Ondrejka). The prime magic square on the right was found by A. W. Johnson, Jr. (Dewdney 1988).According to a 1913 proof of J. N. Muncey (cited in Gardner 1984, pp. 86-87), the smallest magic square composed of consecutive odd primes including..

Magic graph

An edge-magic graph is a labeled graph with graph edges labeled with distinct elements so that the sum of the graph edge labels at each graph vertex is the same.A vertex-magic graph labeled graph vertices which give the same sum along every straight line segment. No magic pentagrams can be formed with the number 1, 2, ..., 10 (Trigg 1960; Langman 1962, pp. 80-83; Dongre 1971; Richards 1975; Buckley and Rubin 1977-1978; Trigg 1998), but 168 almost magic pentagrams (in which the sums are the same for four of the five lines) can. The figure above show a magic pentagram with sums 24 built using the labels 1, 2, 3, 4, 5, 6, 8, 9, 10, and 12 (Madachy 1979).

Rook polynomial

A rook polynomial is a polynomial(1)whose number of ways nonattacking rooks can be arranged on an chessboard. The rook polynomials are given by(2)where is an associated Laguerre polynomial.The first few rook polynomials on square boards are(3)(4)(5)(6)(OEIS A021010).As an illustration, note that the case has two ways to place two rooks (i.e., the rook number ), four ways to place one rook (), and one way to place no rooks (), hence .


The numbers three and four appear prominently in the game of baseball. There are three strikes for an out, three outs per half-inning (i.e., teams switch after three outs, where each first half-inning is called the "top" and the second the "bottom"), innings in a game, giving outs per game (assuming no extra innings). In addition, there are players per team. Four balls are needed for a walk. There are games in the regular season, each for home and away games.The number of bases can either be regarded as three (excluding homeplate) or four (including it).

Evil number

A number in which the first decimal digits of the fractional part sum to 666 is known as an evil number (Pegg and Lomont 2004).However, the term "evil" is also used to denote nonnegative integers that have an even number of 1s in their binary expansions, the first few of which are 0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, ... (OEIS A001969), illustrated above as a binary plot. Numbers that are not evil are then known as odious numbers.Returning to Pegg's definition of evil, the fact that is evil was noted by Keith, while I. Honig (pers. comm., May 9, 2004) noted that the golden ratio is also evil. The following table gives a list of some common evil numbers (Pegg and Lomont 2004).Ramanujan constant 132hard hexagon entropy constant 137139140Stieltjes constant 142pi 144golden ratio 146146151Glaisher-Kinkelin constant 153cube line picking average length155Delian constant 156The probability of the digits of a given real number summing..

Almost integer

An almost integer is a number that is very close to an integer.Surprising examples are given by(1)which equals to within 5 digits and(2)which equals to within 16 digits (M. Trott, pers. comm., Dec. 7, 2004). The first of these comes from the half-angle formula identity(3)where 22 is the numerator of the convergent 22/7 to , so . It therefore follows that any pi approximation gives a near-identity of the form .Another surprising example involving both e andpi is(4)which can also be written as(5)(6)Here, is Gelfond's constant. Applying cosine a few more times gives(7)This curious near-identity was apparently noticed almost simultaneously around 1988 by N. J. A. Sloane, J. H. Conway, and S. Plouffe, but no satisfying explanation as to "why" is true has yet been discovered.Another nested cosine almost integer is given by(8)(P. Rolli, pers. comm., Feb. 19, 2004).An..

Magic hexagon

A magic hexagon of order is an arrangement of close-packed hexagons containing the numbers 1, 2, ..., , where is the th hex number such that the numbers along each straight line add up to the same sum. (Here, the hex numbers are i.e., 1, 7, 19, 37, 61, 91, 127, ...; OEIS A003215). In the above magic hexagon of order , each line (those of lengths 3, 4, and 5) adds up to 38.It was discovered independently by Ernst von Haselberg in 1887 (Bauch 1990, Hemme 1990), W. Radcliffe in 1895 (Tapson 1987, Hemme 1990, Heinz), H. Lulli (Hendricks, Heinz), Martin Kühl in 1940 (Gardner 1963, 1984; Honsberger 1973), Clifford W. Adams, who worked on the problem from 1910 to 1957 (Gardner 1963, 1984; Honsberger 1973), and Vickers (1958; Trigg 1964).This problem and the solution have a long history. Adams came across the problem in 1910. He worked on the problem by trial and error and after many years arrived at the solution which he transmitted to M. Gardner,..

Lights out puzzle

A one-person game played on a rectangular lattice of lamps which can be turned on and off. A move consists of flipping a "switch" inside one of the squares, thereby toggling the on/off state of this and all four vertically and horizontally adjacent squares. Starting from a randomly chosen light pattern, the aim is to turn all the lamps off. The problem of determining if it is possible to start from set of all lights being on to all lights being off is known as the "all-ones problem." As shown by Sutner (1989), this is always possible for a square lattice (Rangel-Mondragon).This can be translated into the following algebraic problem. 1. Each lamp configuration can be viewed as a matrix with entries in (i.e., a (0,1)-matrix, where each 1 represents a burning light and 0 represents a light turned off. For example, for the case,(1)2. The action of the switch placed at can be interpreted as the matrix addition , where is the matrix in which..

Knights problem

The problem of determining how many nonattacking knights can be placed on an chessboard. For , the solution is 32 (illustrated above). In general, the solutions are(1)giving the sequence 1, 4, 5, 8, 13, 18, 25, ... (OEIS A030978,Dudeney 1970, p. 96; Madachy 1979).The minimal number of knights needed to occupy or attack every square on an chessboard (i.e., domination numbers for the knight graphs) are given for , 2, ... by 1, 4, 4, 4, 5, 8, 10, 12, 14, ... (OEIS A006075), with corresponding numbers of such solutions given by 1, 1, 2, 3, 8, 22, 3, ... (OEIS A006076).

Magic number

There are several different kinds of magic numbers. The digital root and magic constant are sometimes known as magic numbers.In baseball, the magic number for a team in first place in a division is the number of games that team must win or the second place team must lose in order to clinch the division. The formula isFor example, the standings for the National League Central Division as of August 9, 2004 are summarized in the following table.teamwinslossesSt. Louis7238Chicago6150Houston5556Cincinnati5457Milwaukee5258Pittsburgh5158There were 162 games in the season so, for example, St. Louis's magic number on that date was

Home plate

Home plate in the game of baseball is an irregular pentagon with two parallel sides, each perpendicular to a base. It seems reasonable to dub such a figure (i.e., a rectangle with a coincident isosceles triangle placed on one side) a "isosceles right pentagon."However, specification of the shape of home plate, illustrated above, as specified by both the Major League Baseball Official Rules and the Little League rulebook (Kreutzer and Kerley 1990) is not physically realizable, since it requires the existence of a (12, 12, 17) right triangle, whereas(Bradley 1996). More specifically, the standards require the existence of an isosceles right triangle with side lengths 8.5 inches and a hypotenuse of length 12 inches, which does not satisfy the Pythagorean theorem.


Bowling, known as "ten pins" throughout most of the world, is a game played by rolling a heavy ball down a long narrow track and attempting to knock down ten pins arranged in the form of a triangle with its vertex oriented towards the bowler. The arrangement of the 10 bowling pins is that of a tetractys and is also triangular number .Up to two balls (or "bowls") are allowed per "frame," and a game consists of ten frames (with a special rule being used for the number of balls awarded in the last frame). If all pins are knocked down on the first ball, the result is called a "strike," no second ball is awarded for that frame (except in the case of a strike being obtained in the tenth and last frame, in which case two extra balls are awarded), and the number of points tallied is 10 plus the number of pins knocked down on the next two balls. If some or none of the pins are knocked down on the first bowl, then a second ball is awarded...

Goat problem

The goat problem (or bull-tethering problem) considers a fenced circular field of radius with a goat (or bull, or other animal) tied to a point on the interior or exterior of the fence by means of a tether of length , and asks for the solution to various problems concerning how much of the field can be grazed.Tieing a goat to a point on the interior of the fence with radius 1 using a chain of length , consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection(1)Taking gives(2)plotted above. Setting (i.e., half of ) leads to the equation(3)which cannot be solved exactly, but which has approximate solution(4)(OEIS A133731).Now instead consider tieing the goat to the exterior of the fence (or equivalently, to the exterior of a silo whose horizontal cross section is a circle) with radius . Assume that , so that the goat is not..

Rubik's cube

Rubik's Cube is a cube in which the 26 subcubes on the outside are internally hinged in such a way that rotation (by a quarter turn in either direction or a half turn) is possible in any plane of cubes. Each of the six sides is painted a distinct color, and the goal of the puzzle is to return the cube to a state in which each side has a single color after it has been randomized by repeated rotations. The puzzle was invented in the 1970s by the Hungarian Ernő Rubik and sold millions of copies worldwide over the next decade.The number of possible positions of Rubik's Cube is(Turner and Gold 1985, Schönert). Hoey showed using the Cauchy-Frobenius Lemma that there are positions up to conjugacy by whole-cube symmetries.The group of operations on Rubik's Cube is known as Rubik's group, and the Cayley graph of that group is called Rubik's graph. The minimum number of turns required to solve the cube from an arbitrary starting position is equal to the graph..

Rubik's clock

Rubik's Clock is a puzzle consisting of 18 small clocks, 14 of which are independent, each of which may be set to any 12-hour position. There are therefore possible configurations in total. The God's number (i.e., the graph diameter of the graph corresponding to Rubik's Clock, which is the minimum number of moves required to solve it from an arbitrary starting position-i.e., in the worst case) was shown by Kogler to be 12 (Kogler 2014; numbers of positions from which the clock can be solved in , 1, ... moves are 1, 330, 51651, 4947912, 317141342, 14054473232, 428862722294, 8621633953202, 101600180118726, 528107928328516, 613251601892918, and 31893880879492, 39248 (A256586;, which sum to as they must.

Railroad track problem

Given a straight segment of track of length , add a small segment so that the track bows into a circular arc. Find the maximum displacement of the bowed track. The Pythagorean theorem gives(1)But is simply , so(2)Solving (1) and (2) for gives(3)Expressing the length of the arc in terms of the centralangle,(4)(5)(6)(7)But is given by(8)so plugging in gives(9)(10)This is a transcendental equation that cannot be solved exactly with a closed-form solution for , but for ,(11)Therefore,(12)(13)Keeping only terms to order ,(14)(15)so(16)and(17)If we take and 1 foot, then feet. Solving equation (◇) numerically, we find that the true answer is feet.

Sultan's dowry problem

A sultan has granted a commoner a chance to marry one of his daughters. The commoner will be presented with the daughters one at a time and, when each daughter is presented, the commoner will be told the daughter's dowry (which is fixed in advance). Upon being presented with a daughter, the commoner must immediately decide whether to accept or reject her (he is not allowed to return to a previously rejected daughter). However, the sultan will allow the marriage to take place only if the commoner picks the daughter with the overall highest dowry. Then what is the commoner's best strategy, assuming he knows nothing about the distribution of dowries (Mosteller 1987)?Since the commoner knows nothing about the distribution of the dowries, the best strategy is to wait until a certain number of daughters have been presented, then pick the highest dowry thereafter (Havil 2003, p. 134). The exact number to skip is determined by the condition that the..


Sudoku (literally, "single number"), sometimes also is a pencil-and-paper logic puzzle whose goal is to complete a grid satisfying various constraints. In the "classic" Sudoku, a square is divided into "regions", with various squares filled with "givens." Valid solutions use each of the numbers 1-9 exactly once within each row, column and region. This kind of sudoku is therefore a particular case of a Latin square.Under the U.S.-only trademarked name "Number Place," Sudoku was first published anonymously by Garns (1979) for Dell Pencil Puzzles. In 1984, the puzzle was used by Nikoli with the Japan-only trademarked name Sudoku (Su = number, Doku = single). Due to the trademark issues, in Japan, the puzzle became well-known as nanpure, or Number Place, often using the English name. Outside Japan, the Japanese name predominates.The puzzle received a large amount of attention in the..

Book stacking problem

How far can a stack of books protrude over the edge of a table without the stack falling over? It turns out that the maximum overhang possible for books (in terms of book lengths) is half the th partial sum of the harmonic series.This is given explicitly by(1)where is a harmonic number. The first few values are(2)(3)(4)(5)(OEIS A001008 and A002805).When considering the stacking of a deck of 52 cards so that maximum overhang occurs, the total amount of overhang achieved after sliding over 51 cards leaving the bottom one fixed is(6)(7)(8)(Derbyshire 2004, p. 6).In order to find the number of stacked books required to obtain book-lengths of overhang, solve the equation for , and take the ceiling function. For , 2, ... book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380, 675214, 4989191, 36865412, 272400600, ... (OEIS A014537) books are needed.When more than one book or card can be used per level, the problem becomes much more complex. For..

St. ives problem

A well-known nursery rhyme states, "As I was going to St. Ives, I met a man with seven wives. Every wife had seven sacks, every sack had seven cats, every cat had seven kitts. Kitts, cats, sacks, wives, how many were going to St. Ives?" Upon being presented with this conundrum, most readers begin furiously adding and multiplying numbers in order to calculate the total quantity of objects mentioned. However, the problem is a trick question. Since the man and his wives, sacks, etc. were met by the narrator on the way to St. Ives, they were in fact leaving--not going to--St. Ives. The number going to St. Ives is therefore "at least one" (the narrator), but might be more since the problem doesn't mention if the narrator is alone.Should a diligent reader nevertheless wish to calculate the sum total of kitts, cats, sacks, wives, plus the man himself, the answer is easily given by the geometric series(1)with..

15 puzzle

The "15 puzzle" is a sliding square puzzle commonly (but incorrectly) attributed to Sam Loyd. However, research by Slocum and Sonneveld (2006) has revealed that Sam Loyd did not invent the 15 puzzle and had nothing to do with promoting or popularizing it. The puzzle craze that was created by the 15 puzzle began in January 1880 in the United States and in April in Europe and ended by July 1880. Loyd first claimed in 1891 that he invented the puzzle, and he continued until his death a 20 year campaign to falsely take credit for the puzzle. The actual inventor was Noyes Chapman, the Postmaster of Canastota, New York, and he applied for a patent in March 1880.The 15 puzzle consists of 15 squares numbered from 1 to 15 that are placed in a box leaving one position out of the 16 empty. The goal is to reposition the squares from a given arbitrary starting arrangement by sliding them one at a time into the configuration shown above. For some initial arrangements,..

Leviathan number

The number , where 666 is the beast number and denotes a factorial. The number has approximately decimal digits.The number of trailing zeros in the Leviathan number is(1)(2)(Pickover 1995).

Legion's numbers

Legion's number of the first kind is defined as(1)(2)where 666 is the beast number. It has 1881 decimaldigits.Legion's number of the second kind is defined as(3)(4)It has approximately digits, and ends with trailing zeros.

Khinchin's constant approximations

Approximations to Khinchin's constant include(1)(2)(3)(4)which are correct to 9, 7, 6, and 5 digits, respectively (M. Hudson, pers. comm., Nov. 20, 2004).

Magic tour

Let a chess piece make a tour on an chessboard whose squares are numbered from 1 to along the path of the chess piece. Then the tour is called a magic tour if the resulting arrangement of numbers is a magic square, and a semimagic tour if the resulting arrangement of numbers is a semimagic square. If the first and last squares traversed are connected by a move, the tour is said to be closed (or "re-entrant"); otherwise it is open. (Note some care with terminology is necessary. For example, Jelliss terms a semimagic tour a "magic tour" and a magic tour a "diagonally magic tour.")Magic knight graph tours are not possible on boards for odd. However, as had long been known, they are possible for all boards of size for . However, the () remained open even since it was first investigated by authors such as Beverley (1848). It was not resolved until an exhaustive computer enumeration of all possibilities was completed on August 5,..

Zebra graph

A zebra graph is a graph formed by all possible moves of a hypothetical chess piece called a "zebra" which moves analogously to a knight except that it is restricted to moves that change by two squares along one axis of the board and three squares along the other. To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable zebra moves are considered edges. The graphs above gives the positions on a square chess boards that are reachable by zebra moves.Square () zebra graphs are connected for .The smallest square board where a tour exists (i.e., for which the underlying zebra graph is Hamiltonian) is the board, first solved in 1886 by Frost (Jelliss). In fact, there are a total of inequivalent (directed) zebra's tours (Hamiltonian cycles) on this board...

Fiveleaper graph

A fiveleaper graph is a graph formed by all possible moves of a hypothetical chess piece called a "fiveleaper" which moves analogously to a knight except that it is restricted to moves that change by three squares along one axis of the board and four squares along the other or by five squares along one axis. To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable fiveleaper moves are considered edges. The fiveleaper gets its name from the fact that all its move have a length of 5 squares.The fiveleaper is similar to the hypothetical chess piece called an "antelope," but it can make an antelope's move or a rook's move of exactly 5 squares.The plots above show the graphs corresponding to antelope graphs on chessboards for to 7.The fiveleaper graph is connected for (trivially) and , Hamiltonian for (trivially) and 8, 10, 12, 14, ...(and all other even but for no odd up to at least ), and traceable..

Rook graph

The rook graph (confusingly called the grid by Brouwer et al. 1989, p. 440) and also sometimes known as a lattice graph (e.g., Bouwer) is the graph Cartesian product of complete graphs, which is equivalent to the line graph of the complete bipartite graph . This is the definition adopted for example by Brualdi and Ryser (1991, p. 153), although restricted to the case . This definition corresponds to the connectivity graph of a rook chess piece (which can move any number of spaces in a straight line-either horizontally or vertically, but not diagonally) on an chessboard.The graph has vertices and edges. It is regular of degree , has diameter 3, girth 3 (for ), and chromatic number . It is also perfect (since it is the line graph of a bipartite graph) and vertex-transitive.The rook graph is also isomorphic to the Latin square graph. The vertices of such a graph are defined as the elements of a Latin square of order , with two vertices being adjacent..

Number guessing

By asking a small number of innocent-sounding questions about an unknown number, it is possible to reconstruct the number with absolute certainty (assuming that the questions are answered correctly). Ball and Coxeter (1987) give a number of sets of questions which can be used.One of the simplest algorithms uses only three queries that can be used to determine an unknown number from an audience member. 1. Ask the person to compute (i.e., three times the secret number ) and announce if the result is even or odd. 2. If you were told that is even, ask the person to compute the number which is half of . If you were told that is odd, ask the person to compute the number which is half of . 3. Ask the person to compute . 4. Ask the person to divide by 9 and to reveal the quotient , discarding any remainder. The original number is then given by if was even, or if was odd. For even, , , , , so . For odd, , , , , so .Another method asks: 1. Multiply the number by 5. 2. Add 6 to the product...


There are many mathematical and recreational problems related to folding. Origami,the Japanese art of paper folding, is one well-known example.It is possible to make a surprising variety of shapes by folding a piece of paper multiple times, making one complete straight cut, then unfolding. For example, a five-pointed star can be produced after four folds (Demaine and Demaine 2004, p. 23), as can a polygonal swan, butterfly, and angelfish (Demaine and Demaine 2004, p. 29). Amazingly, every polygonal shape can be produced this way, as can any disconnected combination of polygonal shapes (Demaine and Demaine 2004, p. 25). Furthermore, algorithms for determining the patterns of folds for a given shape have been devised by Bern et al. (2001) and Demaine et al. (1998, 1999).Wells (1986, p. 37; Wells 1991) and Gurkewitz and Arnstein (2003, pp. 49-59) illustrate the construction of the equilateral triangle, regular..

Poggendorff illusion

The illusion that the two ends of a straight line segment passing behind an obscuring rectangle are offset when, in fact, they are aligned. The Poggendorff illusion was discovered in 1860 by physicist and scholar J. C. Poggendorff, editor of Annalen der Physik und Chemie, after receiving a letter from astronomer F. Zöllner. In his letter, Zöllner described an illusion he noticed on a fabric design in which parallel lines intersected by a pattern of short diagonal lines appear to diverge (Zöllner's illusion). Pondering this illusion, Poggendorff noticed and described another illusion resulting from the apparent misalignment of a diagonal line; an illusion which today bears his name (IllusionWorks).

Impossible fork

The impossible fork (Seckel 2002, p. 151), also known as the devil's pitchfork (Singmaster), blivet, or poiuyt, is a classic impossible figure originally due to Schuster (1964). While each prong of the fork (or, in the original work, "clevis") appears normal, attempting to determine their manner of attachment shows that something is seriously out of whack. The second figure above shows three impossible figures: the ambihelical hexnut in the lower left-hand corner, tribox in the middle, and impossible fork in the upper right.About the time of the impossible fork's discovery by Schuster (1964), it was used by Mad Magazine as a recurring theme. Their term for it was "poiuyt," which corresponds to the third row of a standard keyboard typed from right to left. The "poiuyt" was commonly used in Mad throughout the 1960s indicating absurdity or impossibility.Hayward incorporated this figure into a picture..

Penrose triangle

The Penrose triangle, also called the tribar (Cerf), tri-bar (Ernst 1987), impossible tribar (Pappas 1989, p. 13), or impossible triangle, is an impossible figure published by Penrose and Penrose (1958). Penrose triangles appear prominently in the works of Escher, who not only inspired creation of this object (Escher 1954, Penrose and Penrose 1958), but also subsequently publicized it.The Penrose triangle can be extended to -gonal barred objects (Cerf, Elber), including the so-called tribox.The figure was drawn earlier by artist Oscar Reutersvärd in 1934 during a "long lecture." For this, he was honored with a stamp by the government of Sweden in 1982 (Miller).The Penrose triangle appears on the cover of Raghavachary (2004).Henderson (2006) offers an impossible triangle net...

Penrose stairway

An impossible figure in which a stairway in the shape of a square appears to circulate indefinitely while still possessing normal steps (Penrose and Penrose 1958). The Dutch artist M. C. Escher included a Penrose stairway in his mind-bending illustration "Ascending and Descending" (Bool et al. 1982, p. 321; Forty 2003, Plate 68). Distorted variations of the stairway are also depicted in Escher's "House of Stairs" (Bool et al. 1982, p. 301; Forty 2003, Plate 40).In the 1998 film The Avengers, Uma Thurman is shown walking down a Penrose stairway and ending up back where she began.


The tribox, also called the Penrose rectangle or Penrose square, is an impossible figure that is the generalization of the Penrose triangle from a triangle to a square. Similar -gonal figures can also be constructed for (Elber).The figure above shows three impossible figures: the ambihelical hexnut in the lower left-hand corner, tribox in the middle, and impossible fork in the upper right.

Fraser's spiral

An optical illusion named after British psychologist James Fraser, who first studied the illusion in 1908 (Fraser 1908). The illusion is also known as the false spiral, or by its original name, the twisted cord illusion. While the image appears to be a spiral formed by a rope containing twisted strands of two different colors, it actually consists of concentric circles of twisted cords.The visual distortion is produced by combining a regular line pattern (the circles) with misaligned parts (the differently colored strands). Zöllner's illusion and the café wall illusion are based on a similar principle, like many other visual effects, in which a sequence of tilted elements causes the eye to perceive phantom twists and deviations.

Scintillating grid illusion

In the above illustration, black dots appear to form and vanish at the intersections of the gray horizontal and vertical lines. When focusing attention on a single white dot, some gray dots nearby and some black dots a little further away also seem to appear. More black dots seem to appear as the eye is scanned across the image (as opposed to focusing on a single point). Strangely, the effect seems to be reduced, but not eliminated, when the head is cocked at a angle. The effect seems to exist only at intermediate distances; if the eye is moved very close to or very far away from the figure, the phantom black dots do not appear.The illusion is known as the scintillating grid, and was discovered by E. Lingelbach in 1994. It is a modification of the Hermann grid illusion.

Café wall illusion

The café wall illusion, sometimes also called the Münsterberg illusion (Ashton Raggatt McDougall 2006), is an optical illusion produced by a black and white rectangular tessellation when the tiles are shifted in a zigzag pattern, as illustrated above. While the pattern seems to diverge towards the upper and lower right corners in the upper figure, the gray lines are actually parallel. Interestingly, the illusion greatly diminishes if black lines are used instead of gray.Gregory and Heard (1979) first noticed the illusion on the wall decoration of a café in Bristol, England. The café wall illusion is only one among many visual distortion effects involving parallel lines. The most famous example of this kind is Zöllner's illusion.The image above shows a picture of a building in Melbourne, Australia designed to exhibit this illusion (C. L. Taylor, pers. comm., Aug. 5, 2006). The building,..

Bullseye illusion

Although the inner shaded region has the same area as the outer shaded annulus,it appears to be larger. Since the rings are equally spaced, (1)(2)

Printer's errors

Typesetting "errors" in which exponents or multiplication signs are omitted but the resulting expression is equivalent to the original one. Examples include(1)(2)(3)and(4)where a whole number followed by a fraction is interpreted as a mixed fraction (e.g., ). D. Wilson computed all possible errors obtained by dropping exponents in a product for bases 2 to 15 and numbers :(5)(6)(7)Wilson also gave the numbers and , where the two digit base- satisfies(8)and for which there exist an infinite number of examples.


A cipher is an algorithm that converts data (plaintext) to an obfuscated form that is not directly readable. Ciphers are usually used with the intention of hiding the contents of a message or document from unauthorized persons. Ciphers can also be used to verify identity on the Internet.Cipher algorithms usually require a special "key" that can be used to encrypt the message. Usually, the key provides sufficient information for easy decryption of the ciphertext, however, some ciphers require a different keys for decryption. These algorithms are termed "asymmetric" key ciphers.Most ciphers are reversible, but there do exist algorithms that are non-reversible, termed one-way ciphers, or trap-door algorithms. These are commonly used for comparing passwords: the correct password is encrypted with the algorithm. Passwords that are then entered are encrypted with the same algorithm. The encrypted forms are then..

Caesar's method

Caesar's method is an encryption scheme involving shifting an alphabet (so , , , , etc., ,,). It is one of the most basic encryption methods, and is a specialized form of a transposition cipher.Because the alphabet is rotated, the shift is consistent. A mapping of ,, etc. is termed ROT 1 because the letters are shifted by one. Note that would map to due to the circular aspect of the rotation.Encrypting with ROT and ROT yields the same ciphertext if and only if (mod 26).ROT 13 is a popular encryption method on the Internet since it obfuscates text sufficiently to prevent accidental reading of the text. It is commonly used to list answers of puzzles, for example.Caesar's method is a very weak encryption scheme, since there are 26 possibilities (of which one, ROT 26, is trivial). These can be easily checked by hand...

Partition function p congruences

The fraction of odd values of the partition function P(n) is roughly 50%, independent of , whereas odd values of occur with ever decreasing frequency as becomes large. Kolberg (1959) proved that there are infinitely many even and odd values of .Leibniz noted that is prime for , 3, 4, 5, 6, but not 7. In fact, values of for which is prime are 2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, ... (OEIS A046063), corresponding to 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575). Numbers which cannot be written as a product of are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (OEIS A046064), corresponding to numbers of nonisomorphic Abelian groups which are not possible for any group order.Ramanujan conjectured a number of amazing and unexpected congruences involving . In particular, he proved(1)using Ramanujan's identity (Darling 1919; Hardy and Wright 1979; Drost 1997; Hardy 1999, pp. 87-88; Hirschhorn 1999). Ramanujan (1919) also showed that(2)and..

Butterfly function

The fractal-like two-dimensional functionThe function is named for the appearance of a butterfly-like pattern centered around the origin (left figure). In the above illustration, the left plot runs from to 5 and the right plot runs from to 20.


The randomization of a deck of cards by repeated interleaving. More generally, a shuffle is a rearrangement of the elements in an ordered list. Shuffling by exactly interleaving two halves of a deck is called a riffle shuffle. Shuffling by successively interchanging the cards in position 1, 2, ..., with cards in randomly chosen positions is known as an exchange shuffle. Normal shuffling leaves gaps of different lengths between the two layers of cards and so randomizes the order of the cards.Keller (1995) showed that roughly shuffles are needed just to randomize the bottom card.

Juggler sequence

Define the juggler sequence for a positive integer as the sequence of numbers produced by the iteration(1)where denotes the floor function. For example, the sequence produced starting with the number 77 is 77, 675, 17537, 2322378, 1523, 59436, 243, 3787, 233046, 482, 21, 96, 9, 27, 140, 11, 36, 6, 2, 1.Rather surprisingly, all integers appear to eventually reach 1, a conjecture that holds at least up to (E. W. Weisstein, Jan. 23, 2006). The numbers of steps needed to reach 1 for starting values of , 2, ... are 0, 1, 6, 2, 5, 2, 4, 2, 7, 7, 4, 7, 4, 7, 6, 3, 4, 3, 9, 3, ... (OEIS A007320), plotted above. The high-water marks for numbers of steps are 0, 1, 6, 7, 9, 11, 17, 19, 43, 73, 75, 80, 88, 96, 107, 131, ... (OEIS A095908), which occur for starting values of 1, 2, 3, 9, 19, 25, 37, 77, 163, 193, 1119, ... (OEIS A094679).The smallest integers requiring steps to reach 1 for , 2, ... are 1, 2, 4, 16, 7, 5, 3, 9, 33, 19, 81, 25, 353, ... (OEIS A094670)...


Origami is the Japanese art of paper folding. In traditional origami, constructions are done using a single sheet of colored paper that is often, though not always, square. In modular origami, a number of individual "units," each folded from a single sheet of paper, are combined to form a compound structure. Origami is an extremely rich art form, and constructions for thousands of objects, from dragons to buildings to vegetables have been devised. Many mathematical shapes can also be constructed, especially using modular origami. The images above show a number of modular polyhedral origami, together with an animated crane constructed in the Wolfram Language by L. Zamiatina.To distinguish the two directions in which paper can be folded, the notations illustrated above are conventionally used in origami. A "mountain fold" is a fold in which a peak is formed, whereas a "valley fold" is a fold forming..

Strang's strange figures

Strang's strange figures are the figures produced by plotting a periodic function as a function of an integer argument for , 2, .... Unexpected patterns and periodicities result from near-commensurabilities of certain rational numbers with the period (Richert 1992). Strang figures are shown above for a number of common functions.

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