Impartial games whose rulesets produce given continued fractions
Combinatorics
2013-02-04 v1
Abstract
We study 2-player impartial games of the form take-away which produce P-positions (second player winning positions) corresponding to complementary Beatty sequences, given by the continued fractions (1;k,1,k,1,...) and (k+1;k,1,k,1,...). Our problem is the opposite of the main field of research in this area, which is to, given a game, understand its set of P-positions. We are rather given a set of (candidate) P-positions and look for "simple" rules. Our rules satisfy two criteria, they are given by a closed formula and they are invariant, that is, the available moves do not depend on the position played from (for all options with non-negative coordinates).
Cite
@article{arxiv.1302.0271,
title = {Impartial games whose rulesets produce given continued fractions},
author = {Urban Larsson and Mike Weimerskirch},
journal= {arXiv preprint arXiv:1302.0271},
year = {2013}
}