Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.8

Nim Fractals

Tanya Khovanova
Department of Mathematics
Cambridge, MA 02139

Joshua Xiong
Acton-Boxborough Regional High School
Acton, MA 01719


We enumerate P-positions in the game of Nim in two different ways. In one series of sequences we enumerate them by the maximum number of counters in a pile. In another series of sequences we enumerate them by the total number of counters.

We show that the game of Nim can be viewed as a cellular automaton, where the total number of counters divided by 2 can be considered as a generation in which P-positions are born. We prove that the three-pile Nim sequence enumerated by the total number of counters is a famous toothpick sequence based on the Ulam-Warburton cellular automaton. We introduce 10 new sequences.

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(Concerned with sequences A000007 A000012 A000027 A016945 A048883 A130665 A236305 A237686 A237711 A238147 A238759 A241522 A241523 A241717 A241718 A241731.)

Received May 22 2014; revised version received July 13 2014. Published in Journal of Integer Sequences, July 14 2014.

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