Peg solitaire and Conway's soldiers on infinite graphs
Abstract
Peg solitaire is classically a oneplayer game played on a square grid board containing pegs. The goal of the game is to have a single peg remaining on the board by sequentially jumping a peg over an adjacent peg onto an empty square while eliminating the jumped peg. Conway's soldiers is a related game played on $\mathbb{Z}^2$ with pegs initially located on the halfspace $y \le 0$. The goal is to bring a peg as far as possible on the board using peg solitaire jumps. Conway showed that bringing a peg to the line $y = 5$ is impossible with finitely many jumps. Applying Conway's approach, we prove an analogous impossibility property on graphs. In addition, we generalize peg solitaire on finite graphs as introduced by Beeler and Hoilman (2011) to an infinite game played on countable graphs.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.11128
 Bibcode:
 2021arXiv210911128V
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 05C57 (Primary) 05C63;
 91A43;
 91A46 (Secondary)
 EPrint:
 12 pages, 5 figures