Peg solitaire is traditionally a one-player game played on a grid board filled with 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 half-space $y \le 0$. 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 countably infinite graphs.
翻译：Peg Solitaire 传统上是一个玩家游戏, 玩于一个用钉子填满的网格板上。 游戏的目标是在板上保留一个单钉子。 在清除跳板时, 在空方块上按顺序在相邻的钉子上跳一个钉子, 消灭跳板。 Conway 的士兵是一个相关的游戏, 玩于$\ mathbb ⁇ 2$ 上, 上面的钉子最初位于半空 $y\le 0$ 上。 目标是尽可能地在板上设置一个钉子, 并使用贝索利尔跳跃。 Conway 显示, 将一条钉子绑在一条线上 $y = 5$ 的钉子上, 只能有有限的多次跳动是不可能的。 套用 Conway 的方法, 我们在图形上证明一个类似不可能的属性。 此外, 我们把比勒 和 霍尔曼 (2011 ) 介绍的定数图形上的比莱尔 和霍尔曼 (2011 ) 引入的平面图上, 到一个无限的游戏。
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