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Wednesday, March 19, 2014

**Abstract:** The talk will be targeted at an undergraduate audience, but anyone is welcome! No prior knowledge will be expected.

Peg solitaire is a table game people may have played at Cracker Barrel. Beeler and Hoilman generalized the game to connected graphs. A graph is a collection of vertices and edges. In the graph game, pegs are placed on all but one vertex. If $xyz$ form a 3-vertex path and $x$ and $y$ each have a peg but $z$ does not, then we can remove the pegs at $x$ and $y$ and place a peg at $z$. By analogy with the moves in the original game, this is called a jump. The goal of the peg solitaire game on graphs is to find jumps that reduce the number of pegs on the graph to 1.

Beeler and Rodriguez proposed a variant where we instead want to maximize the number of pegs remaining when no more jumps can be made. Maximizing over all initial locations of a single hole, the maximum number of pegs left on a graph $G$ when no jumps remain is the fool's solitaire number $F(G)$. The join of two graphs $G$ and $H$ consists of disjoint copies of $G$ and $H$ and all possible edges between them. We determine the fool's solitaire number for the join of any graphs $G$ and $H$.