1-Dimensional Peg Solitaire, and Duotaire
Cristopher Moore, David Eppstein · 2002
. We solve the problem of one-dimensional Peg Solitaire. In particular, we show that the set of configurations that can be reduced to a single peg forms a regular language, and that a linear-time algorithm exists for reducing any configuration to the minimum number of pegs. We then look at the impartial two-player game, proposed by Ravikumar, where two players take turns making peg moves, and whichever player is left without a move loses. We calculate some simple nim-values and discuss when the game separates into a disjunctive sum of smaller games. In the version where a series of hops can be made in a single move, we show that neither the P -positions nor the N -positions (i.e. wins for the previous or next player) are described by a regular or context-free language. 1. Solitaire Peg Solitaire is a game for one player. Each move consists of hopping a peg over another one, which is removed. The goal is to reduce the board to a single peg. The best-known forms of the game take place on cross-shaped or triangular boards, and it has been marketed as “Puzzle Pegs” and “Hi-Q.” Discussions and various solutions can be found in [1; 2; 3; 4; 5]. In [6], Guy proposes one-dimensional Peg Solitaire as an open problem in the field of combinatorial games. Here we show that the set of solvable configurations forms a regular language, i.e. it can be recognized by a finite-state automaton. In fact, this was already shown in 1991 by Plambeck ([7], Introduction and Ch.5) and appeared as an exercise in a 1974 book of Manna [8]. More generally, B. Ravikumar showed that the set of solvable configurations on rectangular boards of any finite width is regular [9], although finding an explicit grammar seems to be difficult on boards of width greater than 2. Thus there is little new about this result. However, it seems not to have appeared in print, so here it is.