Fool's solitaire on graphs

Abstract

In recent work by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. Normally, the goal of peg solitaire is to minimize the number of pegs remaining at the end of the game. In this paper, we consider the open problem of determining the maximum number of pegs that can remain at the end of the game, under the restriction that we must jump whenever possible. In this paper, we give bounds for this number. We also determine it exactly for several well-known families of graphs. Several open problems regarding this number are also given.