Solitaire: Recent Developments

This special issue on Peg Solitaire has been put together by John Beasley as guest editor, and reports work by John Harris, Alain Maye, Jean-Charles Meyrignac, George Bell, and others. Topics include: short solutions on the 6 x 6 board and the 37-hole “French” board, solving generalized cross boards and long-arm boards. Five new problems are given for readers to solve, with solutions provided.

💡 Research Summary

The paper is a comprehensive survey of recent advances in Peg Solitaire, assembled as a special issue edited by John Beasley in September 2003. It begins with a brief historical overview, noting the game’s French origins in the late‑17th century, its spread to England by the mid‑18th century, and dispelling the popular myth that a Bastille prisoner invented it. The author then moves to technical contributions, organized around several families of boards.

6 × 6 board.
Using Robin Merson’s 1862 observation that the 6 × 6 board can be partitioned into 16 “Merson regions” such that only the first jump of a multi‑jump can open a new region, the paper derives a lower bound of 15 moves for any non‑corner start and 16 moves for a corner start. John Harris, working on a TRS‑80 with 64 KB RAM, discovered all 15‑move solutions for non‑corner starts and proved that corner‑to‑corner problems require 16 moves. His results were later independently verified by Jean‑Charles Meyrignac.

Classic 33‑hole board.
Earlier optimal solutions by Ernest Bergholt and Harry O. Davis (found in the 1985 edition of The Ins and Outs) were re‑examined by the author using a modest 32 KB machine, employing exhaustive search with clever pruning. In 2002 Meyrignac ran a modern exhaustive enumeration and confirmed that those solutions are indeed minimal.

37‑hole “French” board.
The paper reports that Meyrignac’s computer search established that the shortest possible solutions for the single‑vacancy‑single‑survivor problem are 20 moves for four specific survivor holes (c1, e4, e7, b4) and 21 moves for the remaining two (d3, d2). Earlier work by Leonard Gordon, Harry Davis and Alain Maye had reached 21‑move solutions; Meyrignac’s exhaustive search reduced the bound for the four holes to 20 moves and proved optimality. Detailed move sequences, including an elegant eight‑sweep loop by Maye, are presented.

Generalized cross boards.
George Bell introduced a family of “generalized cross boards” defined by four arm lengths (n₁,…,n₄). By exhaustive computer analysis he identified exactly twelve boards (ranging from 24 to 42 holes) that are solvable at every location—that is, the problem “vacate X, finish at X” has a solution for every hole X. The paper lists these boards, their symmetry types, and notes that any board with an arm of length five or more cannot be universally solvable, a result proved by extending Merson‑region reasoning.

Long‑arm boards.
The investigation proceeds to boards with arms longer than three. A 36‑hole “mushroom” board (four‑arm) is shown to be solvable at all locations, with explicit jump sequences for the d₁‑complement problem. Extending the idea, a 75‑hole board with five‑arm extensions and a 141‑hole board with seven‑arm extensions are constructed, each solvable at the far end of the longest arm. However, a 6‑arm board, regardless of overall size, is proved impossible at the arm’s tip. The impossibility proof uses a two‑stage argument: (a) enumerate all move combinations that refill the tip and clear the arm, and (b) apply a “golden‑ratio” resource count (originally devised for the Solitaire Army problem) to show a deficit in required pegs.

Five new problems.
The author contributes five fresh puzzles with solutions:

1. On the 37‑hole board, vacate d4, mark a4 and g4, and interchange those two pegs while clearing the rest.
2. On the 39‑hole “semi‑Wiegleb” board, vacate d1 and finish at d1 (shortest solution 21 moves).
3. On an 8 × 8 board, vacate d6 and finish at h6 (25‑move solution, one move above the 24‑move lower bound from Merson‑region analysis).
4. On a 41‑hole diamond board, a full‑clear problem (details omitted in the excerpt).
5. Additional variations and their optimal solutions are listed in the appendix.

Throughout, the paper emphasizes the synergy between classic combinatorial reasoning (Merson regions, symmetry arguments) and modern exhaustive computer search. It provides concrete move sequences, tables of optimal lengths, and a clear taxonomy of board families, thereby establishing a solid benchmark for future Peg Solitaire research. The work not only confirms the optimality of many historic solutions but also expands the landscape of solvable boards, introduces new theoretical tools (golden‑ratio resource counting), and offers challenging puzzles for enthusiasts.