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A Generalization of Wythoff's Game*

Published online by Cambridge University Press:  20 November 2018

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W.A. Wythoff [1] in 1907 defined a modification of the game of Nim by the following rules:

  1. (i) there are two players who play alternately;

  2. (ii) initially there are two piles of matches, an arbitrary number in each pile;

  3. (iii) a player may take an arbitrary number of matches from one pile or an equal number from both piles but he must take at least one match;

  4. (iv) the player who takes the last match wins the game.

If, after his move, a player leaves one match in one pile and two in the other he can force a win; for if his opponent takes one match from the pile containing two he can take both remaining matches; and similarly for the other possibilities.

Research Article
Copyright © Canadian Mathematical Society 1959



Excerpt from Master of Science thesis, University of Manitoba, 1959.


1. Wythoff, W.A., A modification of the game of Nim, Nieuw. Archief. voor Viskunde (2), 7 (1907), 199-202.Google Scholar
2. Beatty, S., Amer. Math. Monthly 33 (1926), 159. (problem); solutions, ibid. 34(1927), 159.Google Scholar
3. Skolem, T., Mathematica Scandinavica, 5(1957), 57.Google Scholar
4. Coxeter, H.S.M., The golden section, phyllotaxis and Wythoff's game, Scripta Mathematica 19 (1953), 135-143.Google Scholar