A nd game (or nk game) is a generalization of the combinatorial game tic-tac-toe to higher dimensions.[1][2][3] It is a game played on a nd 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 nd 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 nd game is a symmetric combinatorial game.

There are a total of winning lines in a nd game.[2][6]

For any width n, at some dimension k (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 nd game is symmetric.

See also

References

  1. 1 2 "Mathllaneous" (PDF). Retrieved 16 December 2016.
  2. 1 2 3 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. 1 2 3 4 Golomb, Solomon; Hales, Alfred. "Hypercube Tic-Tac-Toe" (PDF). Archived from the original (PDF) on 29 April 2016. Retrieved 16 December 2016.
  5. 1 2 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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