|
M,n,k-game
An ''m'',''n'',''k''-game is an abstract board game in which two players take turns in placing a stone of their color on an ''m''-by-''n'' board, the winner being the player who first gets ''k'' stones of their own color in a row, horizontally, vertically, or diagonally.J. W. H. M. Uiterwijk and H. J van der Herik, ''The advantage of the initiative'', Information Sciences 122 (1) (2000) 43-58. Jaap van den Herik, Jos W.H.M. Uiterwijk, Jack van Rijswijck (2002). "Games solved: Now and in the future". Artificial Intelligence. Thus, tic-tac-toe is the 3,3,3-game and free-style gomoku is the 15,15,5-game. An ''m'',''n'',''k''-game is also called a ''k''-in-a-row game on an ''m''-by-''n'' board. The ''m'',''n'',''k''-games are mainly of mathematical interest. One seeks to find the game-theoretic value, the result of the game with perfect play. This is known as solving the game. Strategy stealing argument A standard strategy stealing argument from combinatorial game theory shows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pente
Pente is an Abstract strategy game, abstract strategy board game for two or more players, created in 1977 by Gary Gabrel. A member of the M,n,k-game, m,n,k game family, Pente stands out for its Custodian capture, custodial capture mechanic, which allows players to "sandwich" pairs of stones and capture them by flanking them on either side. This changes the overall Tactic (method), tactical assessments players face when compared to pure placement m,n,k games such as Gomoku. Rules Pente is played on a 19x19 grid of intersections similar to a Go board. Players alternate placing stones of their color on empty intersections, with White always assuming the opening move. The goal of the game is to either align five or more stones of the same color in a row in any vertical, horizontal or diagonal direction or to make five captures. Stones are captured by custodial capture (flanking an adjacent pair of an opponent's stones directly on either side with your own stones). Captures consist o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tic-tac-toe 5
Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid, one with Xs and the other with Os. A player wins when they mark all three spaces of a row, column, or diagonal of the grid, whereupon they traditionally draw a line through those three marks to indicate the win. It is a solved game, with a forced draw assuming best play from both players. Names In American English, the game is known as "tic-tac-toe". It may also be spelled "tick-tack-toe", "tick-tat-toe", or "tit-tat-toe". In Commonwealth English (particularly British, South African, Indian, Australian, and New Zealand English), the game is known as "noughts and crosses", alternatively spelled "naughts and crosses". This name derives from the shape of the marks in the game (i.e the X and O); "nought" is another name for the number zero, while "cross" re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in ''n'' dimensions is equal to \sqrt. An ''n''-dimensional hypercube is more commonly referred to as an ''n''-cube or sometimes as an ''n''-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes. The hypercube is the special case of a hyperrectangle (also called an ''n-orthotope''). A ''unit hypercube'' is a hypercube whose side has length one unit. Often, the hypercube whose corners (or ''vertices'') are the 2''n'' points in R''n'' with each coordinate equal to 0 or 1 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nd Game
A ''n''''d'' game (or ''n''''k'' game) is a generalization of the combinatorial game tic-tac-toe to higher dimensions. It is a game played on a ''n''''d'' hypercube with 2 players. If one player creates a line of length ''n'' of their symbol (X or O) they win the game. However, if all ''n''''d'' spaces are filled then the game is a draw. Tic-tac-toe is the game where ''n'' equals 3 and ''d'' equals 2 (3, 2). Qubic is the game. The or games are trivially won by the first player as there is only one space ( and ). A game with and cannot be won if both players are playing well as an opponent's piece will block the one-dimensional line. Game theory An ''nd'' game is a symmetric combinatorial game. There are a total of \frac winning lines in a ''n''''d'' game. For any width ''n'', at some dimension ''d'' (thanks to the Hales-Jewett theorem), there will always be a winning strategy for player X. There will never be a winning strategy for player O because of the Strateg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaplansky's Game
Kaplansky's game or Kaplansky's ''n''-in-a-line is an abstract board game in which two players take turns in placing a stone of their color on an infinite lattice board, the winner being the player who first gets ''k'' stones of their own color on a line which does not have any stones of the opposite color on it. It is named after Irving Kaplansky. General results * ''k'' ≤ 3 is a first-player win. * 4 ≤ ''k'' ≤ 7 is believed to be draw, but this remains unproven. * ''k'' ≥ 8 is a draw: Every player can draw via a "pairing strategy" or other "draw strategy" of ''m'',''n'',''k''-game. See also * ''m'',''n'',''k''-game * Hex (board game) Hex (also called Nash) is a two player abstract strategy board game in which players attempt to connect opposite sides of a rhombus-shaped board made of hexagonal cells. Hex was invented by mathematician and poet Piet Hein in 1942 and later r ... * Harary's generalized tictactoe References {{Tic-Tac-Toe Abstract strategy ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harary's Generalized Tictactoe
Harary's generalized tic-tac-toe or animal tic-tac-toe is a generalization of the game tic-tac-toe, defining the game as a race to complete a particular polyomino (Harary called them "animals") on a grid of squares. It was devised by Frank Harary in March 1977. Harary tic-tac-toe is similar to the m,n,k-games, of which tic-tac-toe and Gomoku are examples; but in tic-tac-toe the first player is trying to complete ''either'' an I-tromino (a horizontal or vertical line of three squares) ''or'' a diagonal line of three corner-connected squares, whereas in Harary's game there is only a single polyomino involved. Categorization of polyominoes by properties of their games Each polyomino corresponds to a generalized tic-tac-toe game: for example, the I-tromino corresponds to the game in which Player 1 is trying to form an I-tromino and Player 2 is trying to stop him. (Note that it is impossible for Player 2 to form an I-tromino before Player 1 does: the strategy-stealing argument applies. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connect6
Connect6 (; Pinyin: liùzǐqí; ;; ) introduced in 2003 by Professor I-Chen Wu at Department of Computer Science and Information Engineering, National Chiao Tung University in Taiwan, is a two-player strategy game similar to Gomoku. Two players, Black and White, alternately place two stones of their own colour, black and white respectively, on empty intersections of a Go-like board, except that Black (the first player) places one stone only for the first move. The one who gets six or more stones in a row (horizontally, vertically or diagonally) first wins the game. Rules The rules of Connect6 are very simple and similar to the traditional game of Gomoku: * Players and stones: There are two players. Black plays first, and White second. Each player plays with an appropriate color of stones, as in Go and Gomoku. * Game board: Connect6 is played on a square board made up of orthogonal lines, with each intersection capable of holding one stone. In theory, the game board can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connect Four
Connect Four (also known as Connect 4, Four Up, Plot Four, Find Four, Captain's Mistress, Four in a Row, Drop Four, and in the Soviet Union, Gravitrips) is a game in which the players choose a color and then take turns dropping colored tokens into a six-row, seven-column vertically suspended grid. The pieces fall straight down, occupying the lowest available space within the column. The objective of the game is to be the first to form a horizontal, vertical, or diagonal line of four of one's own tokens. It is therefore a type of ''m'',''n'',''k''-game (7, 6, 4) with restricted piece placement. Connect Four is a solved game. The first player can always win by playing the right moves. The game was created by Howard Wexler, and first sold under the ''Connect Four'' trademark by Milton Bradley in February 1974. Gameplay A gameplay example (right), shows the first player starting Connect Four by dropping one of their yellow discs into the center column of an empty game board. The t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
