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Winning strategies: the emergence of base 2 in the game of nim

Published online by Cambridge University Press:  22 June 2022

Eric J. Friedman  and
Adam S. Landsberg
Eric J. Friedman
Affiliation:
International Computer Science Institute, and Department of Industrial Engineering and Operations Research, U.C. Berkeley, USA e-mail: ejf@icsi.berkeley.edu
Adam S. Landsberg
Affiliation:
W. M. Keck Science Department, Claremont McKenna, Pitzer and Scripps Colleges, Claremont, CA 91711 USA e-mail: alandsberg@kecksci.claremont.edu
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Extract

Many players know that the secret of winning the game of nim (and other “impartial” combinatorial games) is to write the sizes of the game’s piles in base 2 and then add them together without carry. The proof of this well-known procedure (described below) is both straightforward and convincing. Nonetheless, the procedure still appears magical, as though a rabbit has been pulled out of a hat. Astute students (and frustrated professors) often ask why the winning strategy for such games involves base 2, and not some other base. After all, nothing about the game of nim itself – the game rules, the configuration of the tokens, etc. – provides any hints about the origin of base 2 in this setting. Minimal insight is offered by most published proofs, which themselves tend to either appear almost wizardly in nature (i.e. assume the base-2 method and show that it miraculously solves the problem) or employ combinatorial arguments that supply little abstract intuition (at least to the authors of this article).

Type
Articles
Information
The Mathematical Gazette , Volume 106 , Issue 566 , July 2022 , pp. 212 - 219
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Copyright
© The Authors, 2022 Published by Cambridge University Press on behalf of The Mathematical Association

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