Nim: The Binary Skeleton
Authored by The Academy · May 27, 2026
The syllabus
A syllabus for the Fellow who wishes to trace the arc from a Chinese picking-stones game to the founding theorem of combinatorial game theory, and to encounter the game's haunting afterlife in Resnais's Marienbad.
Reading order
Charles L. Bouton, Annals of Mathematics, Second Series, Vol. 3, No. 1/4, 1901–1902, pp. 35–39
Begin with Bouton's 1902 paper. Five pages. The proof that the nim-sum (bitwise XOR) determines the winner is one of the cleanest results in all of recreational mathematics.
R. P. Sprague, Tohoku Mathematical Journal, 1935; P. M. Grundy, Eureka, 1939
Sprague and Grundy for the generalisation. The theorem that every impartial game is equivalent to a Nim heap transforms Bouton's result from a curiosity into a foundation.
Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy, Academic Press, 1982 (revised edition, A K Peters, 2001–2004)
Berlekamp, Conway, and Guy for the full theory. Winning Ways is the monument built on the Sprague–Grundy foundation; Nim runs through it as a leitmotif.
Dir. Alain Resnais, screenplay Alain Robbe-Grillet, 1961
End with Resnais. The misère Nim game in Marienbad is the most celebrated cultural reception of a mathematical game — and a meditation on what it means to play a game whose outcome is already determined.
Discussion
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