nd game

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A nd game (or nk game) is a generalization of the 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). The game with d=1, n>1 cannot be won because an opponents piece will block the one-dimensional line.[5]

There are a total of ((n + 2)d - nd)/2 winning lines in a nd game.[2][6]

See also[edit]

References[edit]

  1. ^ a b "Mathllaneous" (PDF). Retrieved December 16, 2016. 
  2. ^ a b c Beck, József (2008-03-20). Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press. ISBN 9780521461009. 
  3. ^ Tichy, Robert F.; Schlickewei, Hans Peter; Schmidt, Klaus D. (2008-07-10). Diophantine Approximation: Festschrift for Wolfgang Schmidt. Springer. ISBN 9783211742808. 
  4. ^ a b c d Golomb, Solomon; Hales, Alfred. "Hypercube Tic-Tac-Toe" (PDF). Retrieved December 16, 2016. 
  5. ^ a b Shih, Davis. "A Scientific Study: k-dimensional Tic-Tac-Toe" (PDF). Retrieved December 16, 2016. 
  6. ^ Epstein, Richard A. (2012-12-28). The Theory of Gambling and Statistical Logic. Academic Press. ISBN 9780123978707.