Infinite chess is any variation of the game of chess played on an unbounded chessboard. Versions of infinite chess have been introduced independently by multiple players, chess theorists, and mathematicians, both as a playable game and as a model for theoretical study. It has been found that even though the board is unbounded, there are ways in which a player can win the game in a finite number of moves.
Classical (FIDE) chess is played on an 8×8 board (64 squares). However, the history of chess includes variants of the game played on boards of various sizes. A predecessor game called courier chess was played on a slightly larger 12×8 board (96 squares) in the 12th century, and continued to be played for at least six hundred years. Japanese chess (shogi) has been played historically on boards of various sizes; the largest is taikyoku shōgi ("ultimate chess"). This chess-like game, which dates to the mid 16th century, was played on a 36×36 board (1296 squares). Each player starts with 402 pieces of 209 different types, and a well-played game would require several days of play, possibly requiring each player to make over a thousand moves. [1] [2] [3] [4]
Chess player Jianying Ji was one of many to propose infinite chess, suggesting a setup with the chess pieces in the same relative positions as in classical chess, with knights replaced by nightriders and a rule preventing pieces from travelling too far from opposing pieces. [5] Numerous other chess players, chess theorists, and mathematicians who study game theory have conceived of variations of infinite chess, often with different objectives in mind. Chess players sometimes use the scheme simply to alter the strategy; since chess pieces, and in particular the king, cannot be trapped in corners on an infinite board, new patterns are required to form a checkmate. Theorists conceive of infinite chess variations to expand the theory of chess in general, or as a model to study other mathematical, economic, or game-playing strategies. [6] [7] [8] [9] [10]
For infinite chess, it has been found that the mate-in-n problem is decidable; that is, given a natural number n and a player to move and the positions (such as on ) of a finite number of chess pieces that are uniformly mobile and with constant and linear freedom, there is an algorithm that will answer if there is a forced checkmate in at most n moves. [11] One such algorithm consists of expressing the instance as a sentence in Presburger arithmetic and using the decision procedure for Presburger arithmetic.
The winning-position problem is not known to be decidable. [11] Not only is there a lack of an upper bound on the smallest such n when there is a mate-in-n, there are also positions for which there is a forced mate but no integer n such that there is a mate-in-n. For example, there is a position such that after one rook move by black, the number of moves until black is checkmated will be one more than the distance by which black moved. [12]
Shogi, also known as Japanese chess, is a strategy board game for two players. It is one of the most popular board games in Japan and is in the same family of games as Western chess, chaturanga, xiangqi, Indian chess, and janggi. Shōgi means general's board game.
An evaluation function, also known as a heuristic evaluation function or static evaluation function, is a function used by game-playing computer programs to estimate the value or goodness of a position in a game tree. Most of the time, the value is either a real number or a quantized integer, often in nths of the value of a playing piece such as a stone in go or a pawn in chess, where n may be tenths, hundredths or other convenient fraction, but sometimes, the value is an array of three values in the unit interval, representing the win, draw, and loss percentages of the position.
Minishogi is a modern variant of shogi. The game was invented around 1970 by Shigenobu Kusumoto of Osaka, Japan. The rules are nearly identical to those of standard shogi, with the exception that it is played on a 5x5 board with a reduced number of pieces, and each player's promotion zone consists only of the rank farthest from the player.
Combinatorial game theory measures game complexity in several ways:
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas can be effectively determined. A theory in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership can exist for them.
Crazyhouse is a chess variant in which captured enemy pieces can be reintroduced, or dropped, into the game as one's own. It was derived as a two-player, single-board variant of bughouse chess.
Tai shogi is a large board variant of shogi. The game dates to the 15th century and is based on earlier large-board shogi games. Before the discovery of taikyoku shogi in 1997, tai shogi was believed to be the largest playable chess variant, if not board game, ever. One game may be played over several long sessions and require each player to make over a thousand moves. It was never a popular game; indeed, a single production of six game sets in the early 17th century was a notable event.
Tsume shogi or tsume (詰め) is the Japanese term for a shogi miniature problem in which the goal is to checkmate the opponent's king. Tsume problems usually present a situation that might occur in a shogi game, and the solver must find out how to achieve checkmate. It is similar to a mate-in-n chess problem.
Whale Shogi is a modern variant of shogi. It is not, however, Japanese: it was invented by R. Wayne Schmittberger of the United States in 1981. The game is similar to Judkins shogi, but with more pieces, and the pieces are named after types of whale.
Dai dai shōgi is a large board variant of shogi. The game dates back to the 15th century and is based on the earlier dai shogi. Apart from its size, the major difference is in the range of the pieces and the "promotion by capture" rule. It is the smallest board variant to use this rule.
Yonin shōgi,, is a four-person variant of shogi. It may be played with a dedicated yonin shogi set or with two sets of standard shogi pieces, and is played on a standard sized shogi board.
Sannin shōgi, or in full kokusai sannin shōgi, is a three-person shogi variant invented circa 1930 by Tanigasaki Jisuke and recently revived. It is played on a hexagonal grid of border length 7 with 127 cells. Standard shogi pieces may be used, and the rules for capture, promotion, drops, etc. are mostly similar to standard shogi. While piece movement differs somewhat from standard shogi, especially in the case of the powerful promoted king, the main difference in play is due to the rules for voluntary and mandatory alliance between two of the three players.
Three-player chess is a family of chess variants specially designed for three players. Many variations of three-player chess have been devised. They usually use a non-standard board, for example, a hexagonal or three-sided board that connects the center cells in a special way. The three armies are differentiated usually by color, with White, Black, and Red serving as the most common color combination.
Zillions of Games is a commercial general game playing system developed by Jeff Mallett and Mark Lefler in 1998. The game rules are specified with S-expressions, Zillions rule language. It was designed to handle mostly abstract strategy board games or puzzles. After parsing the rules of the game, the system's artificial intelligence can automatically play one or more players. It treats puzzles as solitaire games and its AI can be used to solve them.
Shogi, like western chess, can be divided into the opening, middle game and endgame, each requiring a different strategy. The opening consists of arranging one's defenses and positioning for attack, the middle game consists of attempting to break through the opposing defenses while maintaining one's own, and the endgame starts when one side's defenses have been compromised.
Shatar and hiashatar are two chess variants played in Mongolia.
Solving chess consists of finding an optimal strategy for the game of chess; that is, one by which one of the players can always force a victory, or either can force a draw. It is also related to more generally solving chess-like games such as Capablanca chess and infinite chess. In a weaker sense, solving chess may refer to proving which one of the three possible outcomes is the result of two perfect players, without necessarily revealing the optimal strategy itself.
The following outline is provided as an overview of and topical guide to chess:
Joel David Hamkins is an American mathematician and philosopher who is the John Cardinal O'Hara Professor of Logic at the University of Notre Dame. He has made contributions in mathematical and philosophical logic, set theory and philosophy of set theory, in computability theory, and in group theory.
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