Peg solitaire

The first evidence of the game can be traced back to the court of Louis XIV, and the specific date of 1697, with an engraving made ten years later by Claude Auguste Berey of Anne de Rohan-Chabot, Princess of Soubise, with the puzzle by her side.

The August 1697 edition of the French literary magazine Mercure galant contains a description of the board, rules and sample problems.

The objective is, making valid moves, to empty the entire board except for a solitary peg in the central hole.

Thus valid moves in each of the four orthogonal directions are: On an English board, the first three moves might be: There are many different solutions to the standard problem, and one notation used to describe them assigns letters to the holes (although numbers may also be used): This mirror image notation is used, amongst other reasons, since on the European board, one set of alternative games is to start with a hole at some position and to end with a single peg in its mirrored position.

On the English board the equivalent alternative games are to start with a hole and end with a peg at the same position.

There is no solution to the European board with the initial hole centrally located, if only orthogonal moves are permitted.

[1] This analysis introduced a notion called pagoda function which is a strong tool to show the infeasibility of a given generalized peg solitaire problem.

One consequence of this analysis is to put a lower bound on the size of possible "inverted position" problems, in which the cells initially occupied are left empty and vice versa.

It can be proved using abstract algebra that there are only 5 fixed board positions where the game can successfully end with one peg.

Note that if you instead think of * as a hole and o as a peg, you can solve the puzzle by following the solution in reverse, starting from the last picture, going towards the first.

Interesting to note is that the shortest way to fail the game is in six moves, and the solution (besides its rotations and reflections) is unique.

In the results below, it has generated all the board positions it really reached starting with the center vacant and finishing in the central hole.

In Cowboy Bebop: The Movie, the main antagonist, Vincent Volaju, spends most of his free time playing peg solitaire.

The vector for his planned Bioterrorism attack, a type of nanobot, is stored in peg solitaire marbles.