Peg solitaire can be analyzed by putting elements of the Klein group on the diagonals of the board so that whenever you make a legal move the product of the elements with pegs is invariant. You can prove that European solitaire is unsolvable (when the central peg is missing) or that if solve the English version and the last peg isn't in the center, your last move was very, very stupid.
Ori Gurel-Gurevich
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