A Combinatorial Approach to the Solitaire Game David AVIS Antoine DEZA Shmuel ONN Summary: A Combinatorial Approach to the Solitaire Game David AVIS Antoine DEZA Shmuel ONN Abstract. 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 peg solitaire 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: valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. In this paper, we recall these necessary conditions and present new developments: the lattice criterion, which generalizes the rule of three; and results on the strongest pagoda functions, the facets of the solitaire cone. 1 Introduction and Basic Definitions 1.1 Introduction Peg solitaire is a peg game for one player which is played on a board containing a number of holes. The most common modern version uses a cross shaped board with 33 holes ­ see Fig. 1 ­ although a 37 hole board is common in France. Computer versions of the game now feature a wide variety of shapes, including rectangles and triangles. Initially the central hole is empty, the others contain pegs. If in some row (column resp.) two consecutive pegs are adjacent to an empty hole in the same row (column respectively), we may make a move by removing the two pegs and placing one peg in the empty hole. The objective of the game is to make moves until Collections: Computer Technologies and Information Sciences