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A n game (or n game) is a generalization of the combinatorial game tic-tac-toe to higher dimensions. It is a game played on a n 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 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 (4, 3) game. The (n > 0, 0) or (1, 1) games are trivially won by the first player as there is only one space (n = 1 and 1 = 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.

Game theory

An n game is a symmetric combinatorial game.

There are a total of ${\displaystyle {\frac {\left(n+2\right)^{d}-n^{d}}{2}}}$ winning lines in a n 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 Strategy-stealing argument since an n game is symmetric.

See also

• Treblecross – Degenerate tic-tac toe variant

References

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

External links

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

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 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

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