# Infinite chess

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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.

## Background

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] [11]

## Decidability of short mates

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 ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$) 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. [12] One such algorithm consists of expressing the instance as a sentence in Presburger arithmetic and using the decision procedure for Presburger arithmetic.

However, the winning-position problem is not known to be decidable. [12] In addition to the lack of an obvious upper bound on the smallest such n when there is a mate-in-n, there could also be positions for which there is a forced mate but no integer n such that there is a mate-in-n. For example, there could be a position such that after one move by black, the number of moves until black gets checkmated will equal the distance by which black moved whichever piece black moved.

## Variations

• Chess on an infinite plane: 76 pieces are played on an unbounded chessboard. The game uses orthodox chess pieces, plus guards, hawks, and chancellors. The absence of borders makes pieces effectively less powerful (as the king and other pieces cannot be trapped in corners), so the added material helps compensate for this. [13]
• Trappist-1: This variation uses the huygens, a chess piece that jumps prime numbers of squares, possibly preventing the game from ever being solved. [14] . This game feature excludes Trappist-1 from the proof that the mate-in-n problem is decidable.

## Related Research Articles

Chess is a two-player strategy board game played on a checkered board with 64 squares arranged in an 8×8 grid. Played by millions of people worldwide, chess is believed to be derived from the Indian game chaturanga sometime before the 7th century. Chaturanga is also the likely ancestor of the East Asian strategy games xiangqi, janggi, and shogi. Chess reached Europe by the 9th century, due to the Umayyad conquest of Hispania. The pieces assumed their current properties in Spain in the late 15th century, and the modern rules were standardized in the 19th century.

Shogi, also known as Japanese chess or the Game of Generals, is a two-player strategy board game that is the Japanese variant of chess. It is the most popular chess variant in Japan. Shōgi means general's board game.

The knight (♘,♞) is a piece in the game of chess and is represented by a horse's head and neck. Each player starts with two knights, which are located between the rooks and bishops.

A fairy chess piece, variant chess piece, unorthodox chess piece, or heterodox chess piece is a chess piece not used in conventional chess but incorporated into certain chess variants and some chess problems. Fairy pieces vary in the way they move. Because of the distributed and uncoordinated nature of unorthodox chess development, the same piece can have different names, and different pieces the same name in various contexts. Almost all are usually symbolised as inverted or rotated icons of the standard pieces in diagrams, and the meanings of these "wildcards" must be defined in each context separately. Pieces invented for use in chess variants rather than problems sometimes instead have special icons designed for them, but with some exceptions, many of these are not used beyond the individual games they were invented for.

Crazyhouse is a chess variant similar to bughouse chess, but with only two players. It effectively incorporates a rule from the game shogi, in which a player can introduce a captured piece back to the chessboard as their own.

Promotion in chess is a rule that requires a pawn that reaches the eighth rank to be replaced by the player's choice of a bishop, knight, rook, or queen of the same color. The new piece replaces the pawn on the same move. The player cannot convert the pawn to another king nor another pawn. The choice of the new piece is not limited to pieces previously captured, thus promotion can result in a player owning, for example, two or more queens despite starting the game with one. Pawn promotion, or the threat of it, often decides the result in an endgame. Since the queen is the most powerful piece, the vast majority of promotions are to a queen. Promotion to a queen is also called queening; promotion to any other piece is referred to as underpromotion.

Microshogi is a modern variant of shogi, with very different rules for promotion, and demotion. Kerry Handscomb of NOST gave it this English name. Although not confirmed, he credits its invention to the late Oyama Yasuharu, a top level shogi player. The game was invented before 1982.

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 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 chess problem.

Hand shogi is a variant of shogi, however it is not Japanese. It was invented in early 1997 by John William Brown of Lewisville, AR, USA. The name “hand” comes from the fact that each player starts the game with most of their pieces in hand and that each round of a match plays like the hand of a card game.

Heian shōgi is a predecessor of modern shogi. Some form of chess almost certainly reached Japan by the 9th century, if not earlier, but the earliest surviving Japanese description of the rules dates from the early 12th century. Unfortunately, this description does not give enough information to actually play the game, but this has not stopped people from attempting to reconstruct this early form of shogi.

Wa shogi is a large board variant of shogi in which all of the pieces are named for animals. It is played either with or without drops.

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.

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.

Shatar and hiashatar are two chess variants played in Mongolia.

Solving chess means finding an optimal strategy for playing chess, i.e. one by which one of the players can always force a victory, or both can force a draw. It also means more generally solving chess-like games, such as infinite chess. According to Zermelo's theorem, a hypothetically determinable optimal strategy does exist for chess and chess-like games.

The following outline is provided as an overview of and topical guide to chess:

Joel David Hamkins is an American mathematician and philosopher based at the University of Oxford. He has made contributions in mathematical and philosophical logic, particularly set theory and the philosophy of set theory, in computability theory, and in group theory.

A chess variant is a game "related to, derived from, or inspired by chess". Such variants can differ from chess in many different ways.

## References

1. boardgamegeek/taikyoku-shogi boardgamegeek/taikyoku-shogi.
2. chessvariants.com/taikyoku-shogi chessvariants.com/taikyoku-shogi.
3. abstractstrategygames/ultimate-battle-chess.html abstractstrategygames/ultimate-battle-chess.
4. history.chess.taishogi history.chess/taishogi.
5. Infinite Chess at The Chess Variant Pages . An infinite chess scheme represented using ASCII characters.
6. "Infinite Chess, PBS Infinite Series" PBS Infinite Series.
7. Evans, C. D. A.; Joel David Hamkins (2013). "Transfinite game values in infinite chess". arXiv:.Cite journal requires |journal= (help)
8. Evans, C. D. A.; Joel David Hamkins; Norman Lewis Perlmutter (2015). "A position in infinite chess with game value ω4". arXiv:.Cite journal requires |journal= (help)
9. Aviezri Fraenkel; D. Lichtenstein (1981), "Computing a perfect strategy for n×n chess requires time exponential in n", J. Combin. Theory Ser. A, 31 (2): 199–214, doi:10.1016/0097-3165(81)90016-9
10. Evans, C. D. A.; Joel David Hamkins (2013). "Transfinite game values in infinite chess". arXiv:.Cite journal requires |journal= (help)
11. "A position in infinite chess with game value w^4" Transfinite game values in infinite chess, January 2017; A position in infinite chess with game value w^4, October 2015; An introduction to the theory of infinite games, with examples from infinite chess, November 2014; The theory of infinite games: how to play infinite chess and win, August 2014; and other academic papers by Joel Hamkins.
12. Brumleve, Dan; Hamkins, Joel David; Schlicht, Philipp (2012). "The Mate-in-n Problem of Infinite Chess is Decidable". How the World Computes. Lecture Notes in Computer Science. 7318. Springer. pp. 78–88. arXiv:. doi:10.1007/978-3-642-30870-3_9. ISBN   978-3-642-30869-7. S2CID   8998263.
13. Chess on an infinite plane game rules.
14. Trappist-1 game rules