# Nim Fractals

- Published in 2014
- Added on

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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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- pages
- 19

### BibTeX entry

@article{Khovanova2014, abstract = {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.}, author = {Khovanova, Tanya and Xiong, Joshua}, month = {may}, pages = 19, title = {Nim Fractals}, url = {http://arxiv.org/abs/1405.5942 http://arxiv.org/pdf/1405.5942v1}, year = 2014, archivePrefix = {arXiv}, eprint = {1405.5942}, primaryClass = {math.CO}, urldate = {2014-06-05}, collections = {Games to play with friends,Combinatorics} }