  Peg solitaire is classically a one-player game played on a grid board
containing pegs. The goal of the game is to have a single peg remaining on the
board by sequentially jumping with a peg over an adjacent peg onto an empty
cell 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 up 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.
