n^d game

A n^d game (or n^k game) is a generalization of the combinatorial game tic-tac-toe to higher dimensions.^[1]^[2]^[3] It is a game played on a n^d hypercube with 2 players.^[1]^[2]^[4]^[5] 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.^[4] Tic-tac-toe is the game where n equals 3 and d equals 2 (3, 2).^[4] Qubic is the (4, 3) game.^[4] The (n > 0, 0) or (1, 1) games are trivially won by the first player as there is only one space (n⁰ = 1 and 1¹ = 1). A game with d = 1 and n > 1 cannot be won if both players are playing well as an opponent's piece will block the one-dimensional line.^[5]

Game theory

Unsolved problem in mathematics:

Given a width of tic-tac-toe board, what is the smallest dimension such that X is guaranteed a winning strategy?

(more unsolved problems in mathematics)

An n^d game is a symmetric combinatorial game.

There are a total of ${\frac {\left(n+2\right)^{d}-n^{d}}{2}}$ winning lines in a n^d game.^[2]^[6]

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 Strategy-stealing argument since an n^d game is symmetric.

References

^ ^a ^b "Mathllaneous" (PDF). Retrieved 16 December 2016.
^ ^a ^b ^c Beck, József (20 March 2008). Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press. ISBN 9780521461009.
^ Tichy, Robert F.; Schlickewei, Hans Peter; Schmidt, Klaus D. (10 July 2008). Diophantine Approximation: Festschrift for Wolfgang Schmidt. Springer. ISBN 9783211742808.
^ ^a ^b ^c ^d Golomb, Solomon; Hales, Alfred. "Hypercube Tic-Tac-Toe" (PDF). Archived from the original (PDF) on 29 April 2016. Retrieved 16 December 2016.
^ ^a ^b Shih, Davis. "A Scientific Study: k-dimensional Tic-Tac-Toe" (PDF). Retrieved 16 December 2016.
^ Epstein, Richard A. (28 December 2012). The Theory of Gambling and Statistical Logic. Academic Press. ISBN 9780123978707.

External links

Higher-Dimensional Tic-Tac-Toe from the PBS Infinite Series on YouTube

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[:0-1] "Mathllaneous" (PDF). Retrieved 16 December 2016.

[:1-2] Beck, József (20 March 2008). Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press. ISBN 9780521461009.

[3] Tichy, Robert F.; Schlickewei, Hans Peter; Schmidt, Klaus D. (10 July 2008). Diophantine Approximation: Festschrift for Wolfgang Schmidt. Springer. ISBN 9783211742808.

[:2-4] Golomb, Solomon; Hales, Alfred. "Hypercube Tic-Tac-Toe" (PDF). Archived from the original (PDF) on 29 April 2016. Retrieved 16 December 2016.

[:3-5] Shih, Davis. "A Scientific Study: k-dimensional Tic-Tac-Toe" (PDF). Retrieved 16 December 2016.

[6] Epstein, Richard A. (28 December 2012). The Theory of Gambling and Statistical Logic. Academic Press. ISBN 9780123978707.

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v t e Tic-tac-toe
Variants	3D tic-tac-toe Gomoku Notakto Treblecross Number Scrabble Order and Chaos Pente Quantum tic-tac-toe Renju SOS Ultimate tic-tac-toe Wild tic-tac-toe
Related concepts	m,n,k-game n^d game Kaplansky's game Harary's generalized tic-tac-toe Hales–Jewett theorem Strategy-stealing argument Futile game Paper-and-pencil game
Similar games	Nine men's morris Score Four Tic-Stac-Toe Gobblet Quarto Three men's morris Nine Holes Achi Connect Four Connect6 OXO Toss Across Pentago

Game theory

See also

References

External links