Title: A Combinatorial Approach to the Solitaire Game

The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. One of the classical problems concerning the peg solitaire game is the feasibility issue. An early tool used to show the infeasibility of various peg games is the rule of three [Suremain de Missery 1841]. In the 1960s the description of the solitaire cone [Boardman and Conway] provides necessary conditions. Namely, valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. We recall these classic necessary conditions and present recent developments including the lattice criterion - which generalizes the rule of three - and some results on the best pagoda functions, that is, the facets of the solitaire cone.
The talk presents the cone criterion (joint-work with David Avis, McGill).