Infinite chess

Short description: Variation of chess

A simple infinite chess scheme (starting position). More complex schemes have the addition of various fairy chess pieces as well as the infinitely large board.

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.

## Contents

* 1 Background
* 2 Decidability of short mates
* 3 See also
* 4 References
* 5 External links

## Background

Taikyoku shōgi (36×36 squares), most likely starting position. The complete rules of this historical game are not conclusively known.

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]

## 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 ℤ×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.[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.[7]

## See also

* List of chess variants
* Fairy chess pieces

## References

1. ↑ Taikyoku Shogi.
2. ↑ Chess Variants: Taikyoku Shogi.
3. ↑ Abstract Strategy Games.
4. ↑ Free History of Chess.
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. ↑ 7.0 7.1 Evans, C. D. A.; Hamkins, Joel David (2013). "Transfinite game values in infinite chess". arXiv:1302.4377 [math.LO].
8. ↑ Evans, C. D. A.; Hamkins, Joel David; Perlmutter, Norman Lewis (2015). "A position in infinite chess with game value ω4". arXiv:1510.08155 [math.LO].
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. ↑ "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.
11. ↑ 11.0 11.1 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. doi:10.1007/978-3-642-30870-3_9. ISBN 978-3-642-30869-7.

## External links

* Infinitechess.org: Online implementation that supports play against an opponent in the same room or against an opponent on the internet.
* Infinite Chess at The Chess Variant Pages
* Infinite Chess • Infinite Series on YouTube
* Mate-in-Omega, The Great Phenomenon of Infinite Chess on YouTube

* v
* t
* e

Chess variants (list)

Orthodox rules|

| Different starting position|

* Chess960
* Displacement chess
* Transcendental chess

|

Different number of pieces|

* Dunsany's chess
* Handicap

Unorthodox board|

* Balbo's game
* Circular chess
* Cross chess
* Cylinder chess
* Double chess
* Grid chess
* Hexagonal chess
* Infinite chess
* Los Alamos chess
* Masonic chess
* Millennium 3D chess
* Minichess
* Rhombic chess
* Spherical chess
* Three-dimensional chess
* Triangular chess
* Trappist-1

Unorthodox
rules with
traditional
pieces|

| Standard
8×8 board|

* Andernach chess
* Atomic chess
* Beirut chess
* Checkless chess
* Circe chess
* Crazyhouse
* Cubic chess
* Dynamo chess
* Extinction chess
* Hostage chess
* Knight relay chess
* Legan chess
* Losing chess
* Madrasi chess
* Monochromatic chess
* Patrol chess
* PlunderChess
* Portal chess
* Progressive chess
* Three-check chess
* Way of the Knight

|

Multimove variants|

* Avalanche chess
* Kung-Fu Chess
* Marseillais chess
* Monster chess
* Progressive chess

Elements of chance|

* Dark chess
* Dice chess
* Knightmare Chess
* Kriegspiel
* Penultima

Unorthodox rules on
an unorthodox board|

* Alice chess
* Apocalypse
* Chad
* Chessence
* Congo
* Diplomat chess
* Dragonfly
* Jeson Mor
* Parallel worlds chess
* Rollerball

Unorthodox
rules using
non-traditional
pieces|

| Standard 8×8 board|

* Baroque
* Berolina
* Chakra
* Chess with different armies
* Duell
* Falcon–hunter chess
* Grasshopper chess
* Pocket mutation chess

|

Empress and/or
Princess (other pieces: Hawk, Leopard, Unicorn, Fortress, Spider etc)|

* Almost chess
* Capablanca chess
* Capablanca random chess
* Embassy chess
* Grand chess
* Janus Chess
* Maharajah and the Sepoys
* Modern chess
* Seirawan chess

Unorthodox boards|

* Chesquerque
* Chess on a really big board
* Dragonchess
* Gess
* Omega Chess
* Stratomic
* Dekle's triangular chess
* 2000 A.D.
* Wildebeest chess
* Wolf chess

Multiplayer|

* Bosworth
* Bughouse chess
* Business chess
* Djambi
* Enochian chess
* Forchess
* Fortress chess
* Four-player chess
* Quatrochess
* Three-man chess
* Three-player chess
* Tri-chess

Inspired games|

* Arimaa
* Jetan
* Martian chess
* Navia Dratp
* The Duke

Chess-related
games|

| Historical|

* Chaturaji
* Courier chess
* Grant Acedrex
* Shatranj
* Short assize
* Tamerlane chess

|

Regional|

* Banqi
* Game of the Three Kingdoms
* Hiashatar
* Janggi
* Jungle
* Ouk-Khmer
* Makruk
* Manchu chess
* Senterej
* Shatar
* Shogi
* Sittuyin
* Xiangqi

Software|

* ChessV
* Fairy-Max
* XBoard
* Zillions of Games

Related|

* Correspondence chess
* Fairy chess
* Fairy chess piece
* Fairy Chess Review
* The Chess Variant Pages

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Infinite chess. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Infinite_chess&oldid=4158499"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template