Kaplansky's game

Kaplansky's game or Kaplansky's n-in-a-line is an abstract board game in which two players take turns in placing a stone of their color on an infinite lattice board, the winner being the player who first gets k stones of their own color on a line which does not have any stones of the opposite color on it.^[1]^[2]^[3]^[4] It is named after Irving Kaplansky.

General results

k ≤ 3 is a first-player win.
4 ≤ k ≤ 7 is believed to be draw, but this remains unproven.
k ≥ 8 is a draw: Every player can draw via a "pairing strategy" or other "draw strategy" of m,n,k-game.

References

^ Beck, József (1982). "On a generalization of Kaplansky's game". Discrete Mathematics. 42 (1): 27–35. doi:10.1016/0012-365X(82)90050-4.
^ Beck, József (2008). Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press. p. 64. ISBN 9780521461009.
^ Kleitman, D.J.; Rothschild, B.L. (1972). "A generalization of Kaplansky's game". Discrete Mathematics. 22 (2): 173–178. doi:10.1016/0012-365X(72)90082-9.
^ András, Pluhár (2004). "The Recycled Kaplansky's Game". Acta Cybernetica. 16.

[1] Beck, József (1982). "On a generalization of Kaplansky's game". Discrete Mathematics. 42 (1): 27–35. doi:10.1016/0012-365X(82)90050-4.

[2] Beck, József (2008). Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press. p. 64. ISBN 9780521461009.

[3] Kleitman, D.J.; Rothschild, B.L. (1972). "A generalization of Kaplansky's game". Discrete Mathematics. 22 (2): 173–178. doi:10.1016/0012-365X(72)90082-9.

[4] András, Pluhár (2004). "The Recycled Kaplansky's Game". Acta Cybernetica. 16.

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[2]

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General results

See also

References