# Games

## Games Topics

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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...

### Checkers

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)

### 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).

### Chessboard

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...

### 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."..

### Maze

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

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.

### Dice

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..

### Craps

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..

### Boxcars

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

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

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

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..

### 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 .

### 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 .

### 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).

### Bowling

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...

### 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..

### Shuffle

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.

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