A tetromino is a geometric shape composed of four squares, connected orthogonally. This, like pentominoes, is a particular type of polyomino; the corresponding polycube, called a tetracube, is a geometric shape composed of four cubes connected orthogonally. A popular use of tetrominoes is in the video game Tetris; the tetrominoes used in the game are the one-sided tetrominoes. Tetrominoes appeared in Zoda's Revenge: StarTropics II but were called tetrads instead. Polyominos are formed by joining unit squares along their edges. A free polyomino is a polyomino considered up to congruence; that is, two free polyominos are the same if there is a combination of translations and reflections that turns one into the other. A free tetromino is a free polyomino made from four squares. There are five free tetrominoes. One-sided tetrominoes are tetrominoes that may be rotated but not reflected, they are used by, are overwhelmingly associated with, the game Tetris. There are seven distinct one-sided tetrominoes.
Of these seven, three have reflectional symmetry, so it does not matter whether they are considered as free tetrominoes or one-sided tetrominoes. These tetrominoes are: I: four blocks in a straight line. O: four blocks in a 2×2 square. T: a row of three blocks with one added below the center; the remaining four tetrominoes exhibit. These four come in two sets of two; each of the members of these sets is the reflection of the other. The "L-polyominos": J: a row of three blocks with one added below the right side. L: a row of three blocks with one added below the left side; the "skew polyominos": S: two stacked horizontal dominoes with the top one offset to the right. Z: two stacked horizontal dominoes with the top one offset to the left; as free tetrominoes, J is equivalent to L, S is equivalent to Z. But in two dimensions and without reflections, it is not possible to transform J into L or S into Z; the fixed tetrominoes allow only translation, not reflection. There are two distinct fixed I-tetrominoes, four J, four L, one O, two S, four T, two Z, for a total of 19 fixed tetrominoes.
Although a complete set of free tetrominoes has a total of 20 squares, they cannot be packed into a rectangle, like hexominoes, whereas a full set of pentominoes can be tiled into four different rectangles. The proof resembles that of the mutilated chessboard problem: A rectangle having 20 squares covered with a checkerboard pattern has 10 each of light and dark squares, but a complete set of free tetrominoes has 11 squares of one shade and 9 of the other. A complete set of one-sided tetrominoes has 28 squares, requiring a rectangle with 14 squares of each shade, but the set has 15 squares of one shade and 13 of the other. By extension, any odd number of complete sets of either type cannot fit in a rectangle. However, a bag including two of each free tetromino, which has a total area of 40 squares, can fit in 4×10 and 5×8 square rectangles: 5×8 rectangle 4×10 rectangle There are many different ways to cover these rectangles; however the 5×8 and the 4×10 rectangles feature distinct properties: The 5×8 rectangle can be covered in 99392 different ways using 2 complete sets of free tetrominoes.
Counting only once the solutions connected by symmetries and assuming that the equal tetrominoes are non-distinguishable the number goes down to 783. There are only 13 fundamental solutions. There are no solutions with right-left symmetry; the 4×10 rectangle can be covered in 57472 different ways. Assuming that the equal tetrominoes are non-distinguishable the number goes down to 449. In this case there are no symmetric solutions. Two sets of one-sided tetrominoes can be fit to a rectangle in more than one way. By repeating these rectangles in a row, any number of complete sets of either type can fit in a rectangle; the corresponding tetracubes from two complete sets of free tetrominoes can fit in 2×4×5 and 2×2×10 boxes: 2×4×5 box layer 1: layer 2 Z Z T t I: l T T T i L Z Z t I: l l l t i L z z t I: o o z z i L L O O I: o o O O i 2×2×10 box layer 1: layer 2 L L L z z Z Z T O O: o o z z Z Z T T T l L I I I I t t t O O: o o i i i i t l l l The name "tetromino" is a combination of the prefix tetra- "four", "domino".
Each of the five free tetrominoes has a corresponding tetracube, the tetromino extruded by one unit. J and L are the same tetracube, as are S and Z, because one may be rotated around an axis parallel to the tetromino's plane to form the other. Three more tetracubes are possible, all created by placing a unit cube on the bent tricube: Right screw: unit cube placed on top of clockwise side. Chiral in 3D. Left screw: unit cube placed on top of anticlockwise side. Chiral in 3D. Branch: unit cube placed on bend. Not chiral in 3D. In 3D, these eight tetracubes can fit in a 4 × 4 × 8 × 2 × 2 box; the following is one of the solutions. D, S and B represent right screw, left screw and branch point, respectively: 4×4×2 box layer 1: layer 2 S T T T: S Z Z B S S T B: Z Z B B O O L D: L L L D O O D D: I I I I 8×2×2 box layer 1: layer 2 D Z Z L O T T T: D L L L O B S S D D Z Z O B T S: I I I I O B B S If chiral pairs are considered as identical, t
Chomp is a two-player strategy game played on a rectangular chocolate bar made up of smaller square blocks. The players take it in turns to choose one block and "eat it", together with those that are below it and to its right; the top left block is "poisoned" and the player. The chocolate-bar formulation of Chomp is due to David Gale, but an equivalent game expressed in terms of choosing divisors of a fixed integer was published earlier by Frederik Schuh. Chomp is a special case of a poset game where the ordered set on which the game is played is a product of total orders with the minimal element removed. Below shows the sequence of moves in a typical game starting with a 3 × 5 bar: Player A must eat the last block and so loses. Note that since it is provable that player A can win when starting from a 3 × 5 bar, at least one of A's moves is a mistake. Chomp belongs to the category of impartial two-player perfect information games, it turns out. This can be shown using a strategy-stealing argument: assume that the second player has a winning strategy against any initial first-player move.
Suppose that the first player takes only the bottom right hand square. By our assumption, the second player has a response to this, but if such a winning response exists, the first player could have played it as his first move and thus forced victory. The second player therefore cannot have a winning strategy. Computers can calculate winning moves for this game on two-dimensional boards of reasonable size. Three-dimensional Chomp has an initial chocolate bar of a cuboid of blocks indexed as. A move is to take a block together with any block all of whose indices are greater or equal to the corresponding index of the chosen block. In the same way Chomp can be generalised to any number of dimensions. Chomp is sometimes described numerically. An initial natural number is given, players alternate choosing positive divisors of the initial number, but may not choose 1 or a multiple of a chosen divisor; this game models n-dimensional Chomp, where the initial natural number has n prime factors and the dimensions of the Chomp board are given by the exponents of the primes in its prime factorization.
Ordinal Chomp is played on an infinite board with some of its dimensions ordinal numbers: for example a 2 × bar. A move is to pick any block and remove all blocks with both indices greater than or equal the corresponding indices of the chosen block; the case of ω × ω × ω Chomp is a notable open problem. More Chomp can be played on any ordered set with a least element. A move is to remove any element along with all larger elements. A player loses by taking the least element. All varieties of Chomp can be played without resorting to poison by using the misère play convention: The player who eats the final chocolate block is not poisoned, but loses by virtue of being the last player; this is identical to the ordinary rule when playing Chomp on its own, but differs when playing the disjunctive sum of Chomp games, where only the last final chocolate block loses. Nim Hackenbush More information about the game A freeware version for windows Play Chomp online All the winning bites for size up to 14
Sprouts is a paper-and-pencil game with significant mathematical properties. It was invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in the early 1960s; the game is played starting with a few spots drawn on a sheet of paper. Players take turns, where each turn consists of drawing a line between two spots and adding a new spot somewhere along the line; the players are constrained by the following rules. The line must not touch or cross itself or any other line; the new spot cannot be placed on top of one of the endpoints of the new line. Thus the new spot splits the line into two shorter lines. No spot may have more than three lines attached to it. For the purposes of this rule, a line from the spot to itself counts as two attached lines and new spots are counted as having two lines attached to them. In so-called normal play, the player who makes the last move wins. In misère play, the player who makes the last move loses. Misère Sprouts is the only misère combinatorial game, played competitively in an organized forum.
The diagram on the right shows a 2-spot game of normal-play Sprouts. After the fourth move, most of the spots are dead–they have three lines attached to them, so they cannot be used as endpoints for a new line. There are two spots. However, it is impossible to make another move, because a line from a live spot to itself would make four attachments, a line from one live spot to the other would cross lines. Therefore, no fifth move is possible, the first player loses. Live spots at the end of the game are called survivors and play a key role in the analysis of Sprouts, it is not evident from the rules of Sprouts that the game always terminates, since the number of spots increase at each move. The correct approach is to consider the number of lives instead of the number of spots. We can show that if the game starts with n spots, it will end in no more than 3n−1 moves and no fewer than 2n moves. In the following proofs, we suppose that a game starts with n spots and lasts for m moves; each spot starts with three lives and each move reduces the total number of lives in the game by one.
So at the end of the game there are 3n−m remaining lives. Each surviving spot has only one life, so there are 3n−m survivors. There must be at least one survivor, namely the spot added in the final move. So 3n−m ≥ 1; this upper bound is the maximum, it can be attained in many ways by ensuring that there is only one survivor at the end of the game. For instance, the game on the right has 3n − 1 moves. At the end of the game each survivor has two dead neighbors, in a technical sense of "neighbor", different from the ordinary graph notion of adjacency. No dead spot can be the neighbor of two different survivors, for otherwise there would be a move joining the survivors. All other dead spots are called pharisees. Suppose there are p pharisees. N + m = 3 n − m + 2 + p since initial spots + moves = total spots at end of game = survivors + neighbors + pharisees. Rearranging gives: m = 2 n + p / 4 So a game lasts for at least 2n moves, the number of pharisees is divisible by 4; this lower bound on the length of a game is the minimum.
The diagram on the right shows a completed game of 2n moves. It has 2n neighbors and 0 pharisees. Real games seem to turn into a battle over whether the number of moves will be k or k+1 with other possibilities being quite unlikely. One player tries to create enclosed regions containing survivors and the other tries to create pharisees. Since Sprouts is a finite game where no draw is possible, a perfect strategy exists either for the first or the second player, depending on the number of initial spots; the main question about a given starting position is to determine which player can force a win if he or she plays perfectly. When the winning strategy is for the first player, it is said that the outcome of the position is a "win", when the winning strategy is for the second player, it is said that the outcome of the position is a "loss"; the outcome is determined by developing the game tree of the starting position. This can be done by hand only for a small number of spots, all the new results since 1990 have been obtained by extensive search with computers.
Winning Ways for your Mathematical Plays reports that the 6-spot normal game was proved to be a win for the second player by Denis Mollison, with a hand-made analysis of 47 pages. It stood as the record for a long time, until the first computer analysis, done at Carnegie Mellon University, in 1990, by David Applegate, Guy Jacobson, Daniel Sleator, they reached up to 11 spots with some of the best hardware available at the time. Applegate and Sleator observed a pattern in their results, conjectured that the first playe
English draughts or checkers called American checkers or straight checkers, is a form of the strategy board game draughts. It is played on an 8×8 chequered board with 12 pieces per side; the pieces move and capture diagonally forward, until they reach the opposite end of the board, when they are crowned and can thereafter move and capture both backward and forward. As in all forms of draughts, English draughts is played by two opponents, alternating turns on opposite sides of the board; the pieces are red, or white. Enemy pieces are captured by jumping over them; the 8×8 variant of draughts was weakly solved in 2007 by the team of Canadian computer scientist Jonathan Schaeffer. From the standard starting position, both players can guarantee a draw with perfect play. Though pieces are traditionally made of wood, now many are made of plastic, though other materials may be used. Pieces are flat and cylindrical, they are invariably split into one lighter colour. Traditionally and in tournaments, these colours are red and white, but black and red are common in the United States, as well as dark- and light-stained wooden pieces.
The darker-coloured side is referred to as "Black". There are two classes of pieces: kings. Men are single pieces. Kings consist of two men of the same colour, stacked one on top of the other; the bottom piece is referred to as crowned. Some sets have pieces with a crown molded, engraved or painted on one side, allowing the player to turn the piece over or to place the crown-side up on the crowned man, further differentiating kings from men. Pieces are manufactured with indentations to aid stacking; each player starts with 12 men on the dark squares of the three rows closest to that player's side. The row closest to each player is crownhead; the player with the darker-coloured pieces moves first. Turns alternate. There are two different ways to move in English draughts: Simple move: A simple move consists of moving a piece one square diagonally to an adjacent unoccupied dark square. Uncrowned pieces can move diagonally forward only. Jump: A jump consists of moving a piece, diagonally adjacent an opponent's piece, to an empty square beyond it in the same direction.
Men can jump diagonally forward only. A jumped piece is removed from the game. Any piece, king or man, can jump a king. Multiple jumps are possible, if after one jump, another piece is eligible to be jumped—even if that jump is in a different diagonal direction. If more than one multi-jump is available, the player can choose which piece to jump with, which sequence of jumps to make; the sequence chosen is not required to be the one. Jumping is always mandatory: if a player has the option to jump, he must take it if doing so results in disadvantage for the jumping player. For example, a mandated single jump might set up the player such that the opponent has a multi-jump in reply. If a man moves into the kings row on the opponent's side of the board, it is crowned as a king and gains the ability to move both forward and backward. If a man jumps into the kings row, the current move terminates. A player wins by capturing all of the opponent's pieces or by leaving the opponent with no legal move; the game ends by agreement.
In tournament English draughts, a variation called. The first three moves are drawn at random from a set of accepted openings. Two games are played with each player having a turn at either side; this can make for more exciting matches. Three-move restriction has been played in the U. S. championship since 1934. A two-move restriction was used from 1900 until 1934 in the United States and in the British Isles until the 1950s. Before 1900, championships were played without restriction, a style is called Go. One rule of long standing that has fallen out of favour is the huffing rule. In this variation jumping is not mandatory, but if a player does not take their jump, the piece that could have made the jump is blown or huffed, i.e. removed from the board. After huffing the offending piece, the opponent takes their turn as normal. Huffing has been abolished by both the American Checker Federation and the English Draughts Association. Two common rule variants, not recognised by player associations, are:Capturing with a king precedes capturing with a man.
In this case, any available capture can be made at the player's choice. A man that has jumped to become a king, can in the same turn continue to capture other pieces in a multi-jump. There is a standardised notation for recording games. All 32 reachable board squares are numbered in sequence; the numbering starts in Black's double-corner. Black's squares on the first rank are numbered 1 to 4. Moves are recorded as "from-to", so a move from 9 to 14 would be recorded 9-14. Captures are notated with an "x" connecting the end squares; the game result is abbreviated as BW/RW or WW. White resigned after Black's 46th move. [Event "1981 World Championship
There are many variations of the simple rules of Go. Some are ancient digressions, they are side events at tournaments, for example, the U. S. Go Congress holds a "Crazy Go" event every year; the difficulty in defining the rules of Go has led to the creation of many subtly different rulesets. They vary in areas like scoring method, ko, handicap placement, how neutral points are dealt with at the end; these differences are small enough to maintain the character and strategy of the game, are not considered variants. Different rulesets are explained in Rules of Go. In some of the examples below, the effects of rule differences on actual play are minor, but the tactical consequences are substantial. Tibetan Go is played on a 17×17 board, starts with six stones from each color placed on the third line as shown. White makes the first move. There is a unique ko rule: a stone may not be played at an intersection where the opponent has just removed a stone; this ko rule is so different from other major rulesets that it alone changes the character of the game.
For instance, snapbacks must be delayed by at least one move, allowing an opponent the chance to create life. A player who occupies or surrounds all four corner points receives a bonus of 40 points, plus another 10 if the player controls the center point. Sunjang baduk is a different form of Go, it has been played since at least the 7th century. Its most distinctive feature is the prescribed opening; the starting position dictates the placement of 16 stones as shown, the first move is prescribed for Black at the center of the board. At the end of the game, stones inside friendly territory, which are irrelevant to boundary definition, are removed before counting territory. In another Korean variant, the players wager on the outcome of the game. A fixed stake is paid for every ten points on the board. Batoo is a modern Korean variant; the name stems from a combination of the Korean words juntoo. It is played in cyberspace, differs from standard Go in a number of ways, most noticeably in the way in which certain areas of the board are worth different points values.
The other principal difference is that both players place three stones before the game begins, may place a special “hidden stone”, which affects the board as a regular stone but is invisible to the opponent. Batoo became a short-lived fad among young people in Korea around 2011; the first player to capture a stone wins. It was invented by Japanese professional Yasutoshi Yasuda, who describes it in his book Go As Communication. Yasuda was inspired by the need for a medium to address the problem of bullying in Japan, but soon found that "First Capture" works as an activity for senior citizens and developmentally delayed individuals, he sees it as a game in its own right, not just as a prelude to Go, but as a way to introduce simple concepts that lead to Go. For the latter purpose, he recommends progressing to "Most Capture", in which the player capturing the most stones wins; this variation is called Atari Go in the West, where it is becoming popular as a preliminary means of introducing Go itself to beginners, afterward, it is natural to introduce the idea of capturing territory, not just the opponent's stones.
In Miai-Go, each player plays two moves at once, their opponent decides which of the two should stay on the board. In Stoical Go, invented by abstract game designer Luis Bolaños Mures, standard ko rules don't apply. Instead, it's illegal to make a capture. All other rules are the same as in Go. Suicide of one or more stones is not allowed, area scoring is used. All known forced; the nature of the rule itself suggests that forced cycles are either impossible or astronomically rarer than they are in Go when the superko rule is not used. Ko fights proceed in a similar manner to those of Go, with the difference that captures and moves answered by captures aren't valid ko threats. Although snapbacks are not possible in the basic variant, they can be explicitly allowed with an extra rule while retaining the property that all known forced cycles are impossible. Environmental Go called Coupon Go, invented by Elwyn Berlekamp, adds an element of mathematical precision to the game by compelling players to make quantitative decisions.
In lieu of playing a stone, a player may take the highest remaining card from a pack of cards valued in steps of ½ from ½ to 20: the player's score will be the territory captured, plus the total value of cards taken. In effect, the players participate in a downward auction for the number of points they think sente is worth at each stage in the game; the professional players Jiang Zhujiu and Rui Naiwei played the first Environmental Go game in April 1998. Since the variant has seen little activity on the international scene; each player begins the game with a decided upon amount of time. At the end of the game, when the score is counted, the number of seconds remaining on each player's clock is added to their respective score. In Cards Go players draw from a pack of cards contain instructions to play one of a fixed set of occurring shapes. If the said shape cannot be placed on the board an illegal move is deemed to have been played, which necessitates resignation. In Multi-player Go, stones of different colors are used so that three or more players can play together.
The rules must be somewhat altered to create balance in power
A chess variant is a game "related to, derived from, or inspired by chess". Such variants can differ from chess in many different ways, ranging from minor modifications to the rules, to games which have only a slight resemblance. "International" or "Western" chess itself is one of a family of games which have related origins and could be considered variants of each other. Chess is theorised to have been developed from chaturanga, from which other members of this family, such as shatranj and xiangqi evolved. Many chess variants are designed to be played with the equipment of regular chess. Although most variants have a similar public-domain status as their parent game, some have been made into commercial, proprietary games. Just as in traditional chess, chess variants can be played over-the-board, by correspondence, or by computer; some internet chess servers facilitate the play of some variants in addition to orthodox chess. In the context of chess problems, chess variants are called fairy chess.
Fairy chess variants tend to be created for problem composition rather than actual play. There are thousands of known chess variants; the Classified Encyclopedia of Chess Variants catalogues around two thousand, with the preface noting that — with creating a chess variant being trivial — many were considered insufficiently notable for inclusion. The origins of the chess family of games can be traced to the game of chaturanga during the time of the Gupta Empire in India. Over time, as the game spread geographically, modified versions of the rules became popular in different regions. In Sassanid Persia, a modified form became known as shatranj. Modifications made to this game in Europe resulted in the modern game. Courier chess was a popular variant in medieval Europe, which had a significant impact on the "main" variant's development. Other games in the chess family, such as shogi, xiangqi, are developments from chaturanga made in other regions; these related games are considered chess variants, though the majority of variants are, modifications of chess.
The basic rules of chess were not standardised until the 19th century, the history of chess prior to this involves many variants, with the most popular modifications spreading and forming the modern game. While some regional variants have historical origins comparable to or older than chess, the majority of variants are express attempts by individuals or small groups to create new games with chess as a starting point. In most cases the creators are attempting to create new games of interest to chess enthusiasts or a wider audience. Variants have the same public domain status as chess, though a few are proprietary, the materials for play are released as commercial products; the variations from chess may be done to address a perceived issue with the standard game. For example, Chess960, which randomises the starting positions, was invented by Bobby Fischer to combat what he perceived to be the detrimental dominance of opening preparation in chess. Several variants introduce complications to the standard game, providing an additional challenge for experienced players, for example in Kriegspiel, where players cannot see the pieces of their opponent.
A handful, such as No Stress Chess, attempt to simplify the game, so as to be attractive to chess beginners. The table below details some, but not all, of the ways in which variants can differ from the orthodox game: Variants can themselves be developed into further sub-variants, for example Horde chess is a variation upon Dunsany's Chess; some variations are created for the purpose of composing interesting puzzles, rather than being intended for full games. This field of composition is known as fairy chess. Fairy chess gave rise to the term "fairy chess piece", used more broadly across writings about chess variants to describe chess pieces with movement rules other than those of the standard chess pieces. Forms of standardised notation have been devised to systematically describe the movement of these. A distinguishing feature of several chess variants is the presence of one or more fairy pieces. Physical models of common fairy pieces are sold by major chess set suppliers. Individuals notable for creating multiple chess variants include V. R. Parton, Ralph Betza, Philip M. Cohen and George R. Dekle Sr.
Some board game designers, notable for works across a wider range of board games, have attempted to create chess variants. These include Andy Looney. Several chess masters have developed variants, such as Chess960 by Bobby Fischer, Capablanca Chess by José Raúl Capablanca, Seirawan chess by Yasser Seirawan. While chess and xiangqi have professional circuits as well as many organised tournaments for amateurs, play of the majority of chess variants is predominately on a casual basis; some variants have had significant tournaments. Several Gliński's hexagonal chess tournaments were played at the height of the variant's popularity in the 1970s and 1980s. Chess960 has been the subject of tournaments, including in 2018 an "unofficial world championship" between reigning World Chess Champion Magnus Carlsen and fellow high-ranking Grandmaster Hikaru Nakamura. Several internet chess servers facilitate live play of popular variants, including Chess.com and the Free Internet Chess Server. The software packages Zillions of Games and Fairy-Max have been programmed to support many chess variants.
Play in most chess variants is sufficiently similar to chess that games can be recorded with algebraic notation, although additions to this are required. For example, the third dimension in Millennium 3D Chess means that move notatio
The 24 Game is an arithmetical card game in which the objective is to find a way to manipulate four integers so that the end result is 24. For example, for the card with the numbers 4, 7, 8, 8, a possible solution is × 4 = 24; the game has been played in Shanghai since the 1960s. It has been known by other names, including Maths24, but these products are not associated with the copyrighted versions of the 24® Game; the original version of 24 is played with an ordinary deck of playing cards with all the face cards removed. The aces are taken to have the value 1 and the basic game proceeds by having 4 cards dealt and the first player that can achieve the number 24 using only allowed operations wins the hand; some advanced players allow exponentiation, roots and other operations. For short games of 24, once a hand is won, the cards go to the player. If everyone gives up, the cards are shuffled back into the deck; the game ends when the deck is exhausted, the player with the most cards wins. Longer games of 24 proceed by first dealing the cards out to the players, each of whom contributes to each set of cards exposed.
A player who solves a set replenishes their pile, after the fashion of War. Players are eliminated. A different version includes the face cards, Jack and King, giving them the values 11, 12, 13, respectively. 24® Game is a competitive, arithmetical card game aimed predominantly at primary and high school pupils. Although it can be played informally, the game was organised and operated within Southern Africa in a series of interschool, geographically increasing tournaments; the game experienced its peak during the 1990s, is now no longer produced or played in any official manner. The modern version of the 24® Game was invented and copyrighted by engineer and inventor Robert Sun in 1988; the first edition of the game, Single Digits, began appearing in schools in 1989. The 24® game is copyrighted, is produced by Pennsylvania-based Suntex International Inc. Eight other editions of the game would follow; the project aimed to introduce pupils and encourage their participation in mathematics via entertaining activities including the 24® Game as a mathematics teaching tool.
The cards are double-sided, thin cardboard squares with sides measuring 10 cm. The conventional cards bear the Old Mutual logo in the centres of each side, however with the green and white inverted on one side to differentiate. Variations of the cards bore red and black logos; the conventional card displays four numbers, each a single digit from 1 to 9. Numbers may repeat; the cards are designed to be viewed from any angle. The card difficulty is ranked by displaying one, two or three dots in each corner of the card in white and yellow as the difficulty increases. Although official rules were published, the game evolved with common basic rules, many smaller variations. Any number of competitors sit around a table; the cards are placed, in the centre of the table. The first person to cover the card with their hand and claim to have the solution would be given the first opportunity to give their answer; the 24® Game game was intended and played in tournament scenarios, ranging from school to international levels.
Competitors are distributed into tables of four, each with their own adjudicator. The game is played with participants competing for points. Points are earned by solving cards, with one, two or three points assigned to cards of increasing difficulty. After claiming a solution to a card, if a participant failed to give a correct or legal solution their points would be deducted according to the difficulty. Rounds continue for a predefined length of time, at the end of which the points are tallied and the winners proceed. At the top competition levels, there would be no delay between a card's placement and a participant's claiming it; because there are only a finite number of cards that can be made, participants can give solutions from memory. Participants could exploit the small delay between claiming a card and giving its solution, to work out the solution; because of the finite extent of the basic cards, many variations and adaptations of the game were introduced to add complexity to tournaments.
Using the regular cards, the target total could be changed for a round, from the usual 24, to 25, 36 or 48. Instead of many participants competing to solve communal cards for a set time, each participant could be presented with an entire pack of cards; the times taken to work through the pack, solving all cards, would be compared. Early versions go the boxed games contained a cardboard sleeve that could hold two cards side-by-side could be used in the game; the sleeve allows three numbers from each card to be visible, while concealing the fourth number from each. Participants would be required to solve a sleeve by identifying a number that could be common to both cards, as well as giving both card solutions; these "Puzzle Sleeves" are no longer produced, but an edition titled "Variables" has been created to allow the same style of play. New cards were introduced to the games; the first card adaptation was the introduction of fractions. Cards were printed with x and y variables that could appear in many forms, including fractions and algebraic expressions.
Participants would be required to find positive integers less than ten that the variables could represent, as w