Nim Fractals
Combinatorics
2014-05-26 v1
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.
Cite
@article{arxiv.1405.5942,
title = {Nim Fractals},
author = {Tanya Khovanova and Joshua Xiong},
journal= {arXiv preprint arXiv:1405.5942},
year = {2014}
}
Comments
19 pages, 2 figures