9x9 Peg Solitaire
Last Modified August 24, 2005
Copyright © 2005 by George I. Bell
Extended version of this page [Not available online]

This web page documents some interesting properties and solutions for peg solitaire on the square 9x9 board.

Table of Contents:

It is unfortunate that a standard chess board is just slightly too small to play 9x9 peg solitaire on. You can play the game on a go board, however the best board is really a computer. On a computer game you can easily backtrack and record move sequences, and taking the complement of a board position is trivial once a suitable button has been programmed. Many of the solutions below were discovered by hand using a Javascript program which I modified from a version by JC Meyrignac. Unfortunately this game is rather hacked up and I don't think anyone else would find it useful if they can't program in Javascript.

The Central Game

Below is a diagram of a 45 move solution to the central game which can be generalized to larger square and rectangular boards.

Shortest Solution

What is the least number of moves the central game can be solved in? This question is quite difficult to answer computationally.

In 1962, Robin Merson found an elegant argument which gives a lower bound for the length of a solution on the 6x6 board. This argument can be generalized to any square (or even rectangular) null class board, but on an n x n board the bound is not very tight if n is odd. The complement problem from a corner must use at least (n/2+1)² moves, where when n is odd one must round n/2 down to the nearest integer. If the problem does not begin in a corner the bound is one less. On the 6x6 board this gives a lower bound of 15 for all problems that do not begin at a corner. In fact one can come up with solutions in 15 moves, so one immediately has a proof that they are optimal.

On the 9x9 board the lower bound indicates that any solution must have at least 24 moves. The best solution I have seen was constructed by Alain Maye, by hand, and has 34 moves. It is likely it can be done in fewer than 34 moves, but I doubt the minimum length solution is under 28 moves.

Symmetry

On the standard 33-hole board, it has been shown that no solution to the central game can pass through a position with rotational symmetry. This is also true for Weigleb's Board, but not for the 9x9 board. Once one discovers this is possible, how many positions of symmetry could a solution to the central game pass through?

The solution below answers this question. After 8 jumps (or 6 moves) the board position becomes square symmetric (shown in red). The next 60 jumps come in sets of 4 moves that are rotational copies of one another, so every 4 moves you pass through a position with rotational symmetry (shown in green). Then the final 11 jumps (or 6 moves) finish at the center. The final solution has 8+60+11=79 jumps (or 72 moves) and passes through 16 positions with rotational symmetry, 5 of them being square symmetric. It is not possible to go through more than 16 positions with rotational symmetry, because one cannot be reached in under 8 jumps from either end.

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