Nim Fractals

• Published in 2014
• Added on 2014-06-05

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.

Links

• http://arxiv.org/abs/1405.5942
• http://arxiv.org/pdf/1405.5942v1

Other information

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}
}