Peg solitaire

Peg Solitaire is a solitaire-type puzzle game for one player involving movement of pegs on a board with holes. The game is known just as Solitaire in England where the card games are called Patience. Some sets use marbles in a board with indentations.

There are two traditional boards:

    English          European
     · · ·             · · ·
     · · ·           · · · · ·
 · · · · · · ·     · · · · · · ·
 · · · · · · ·     · · · · · · ·
 · · · · · · ·     · · · · · · ·
     · · ·           · · · · ·
     · · ·             · · ·

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

A valid move is to jump a peg orthogonally over an adjacent peg into a hole two positions away and then to remove the jumped peg.

In the diagrams which follow, * is used to indicate a peg (in a hole) and · indicates an empty hole.

Thus valid moves in each of the four orthogonal directions are:

 * * ·  ->  · · *  Jump to right

 · * *  ->  * · ·  Jump to left 

 *      ·
 *  ->  ·  Jump down
 ·      *

 ·      *
 *  ->  ·  Jump up
 *      ·

On an English board, the first three moves might be:

     * * *             * * *             * * *             * * *        
     * * *             * · *             * · *             * * *        
 * * * * * * *     * * * · * * *     * · · * * * *     * · · · * * *    
 * * * · * * *     * * * * * * *     * * * * * * *     * * * · * * *    
 * * * * * * *     * * * * * * *     * * * * * * *     * * * * * * *    
     * * *             * * *             * * *             * * *        
     * * *             * * *             * * *             * * *        

It is very easy to go wrong and find you have two or three widely spaced lone pegs. Many people never manage to solve the problem. Warning: Wikipedia contains spoilers, this article includes one solution to the English standard game..

There are many different solutions to the standard problem, and one notation used to describe them assigns letters to the holes:

    English          European
     a b c             a b c
     d e f           y d e f z
 g h i j k l m     g h i j k l m
 n o p x P O N     n o p x P O N
 M L K J I H G     M L K J I H G
     F E D           Z F E D Y
     C B A             C B A

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.

One solution of the standard English game is:

1. e-x
2. l-j
3. c-k
4. P-f
5. D-P
6. G-I
7. J-H
8. m-G-I
9. i-k
10. g-i
11. L-J-H-l-j-h
12. C-K
13. p-F
14. A-C-K
15. M-g-i
16. a-c-k-I
17. d-p-F-D-P-p
18. o-x

A tactic that can be used is to divide the board into packages of three and to purge (remove) them entirely using one extra peg, the catalyst, that jumps out and then jumps back again. In the example below, the * is the catalyst.:

 * * ·      · · *      · * *      * · ·
   *    ->    *    ->    ·    ->    ·
   *          *          ·          ·

This technique can be used with a line of 3, a block of 2*3 and a 6-peg L shape with a base of length 3 and upright of length 4.

Other alternate games include starting with two empty holes and finishing with two pegs in those holes. Also staring with one hole here and ending with one peg there. On a English board, the hole can be anywhere and the final peg can only end up where multiples of both two and three permit. Thus a hole at a can only leave a single peg at a or C.

A thorough analysis of the game is provided in Winning Ways[?] ISBN 01120911027 in the UK and ISBN 156881142 in the US.

Peg solitaire has been played on other size boards, although the two given above are the most popular. It has also been played on a triangualr board, with jumps allowed in all 3 directions. As long as the variant has the proper "parity" and is large enough, it will probably be solvable.

Brute force attack on standard English peg solitaire

The only place it is possible to end up with a solitary peg, is the centre, or the middle of one of the edges; on the last jump, there will always be an option of choosing whether to end in the centre or the edge.

Following is a table over the number (Possible Board Positions) of possible board positions after n jumps, and the number (No Further Jumps) of those board positions, from which no further jumps are possible.

If one board position can be rotated and/or flipped into another board position, the board positions are counted as identical.

n PBP NFJ 1 1 0 2 2 0 3 8 0 4 39 0 5 171 0 6 719 1 7 2757 0 8 9751 0 9 31312 0 10 89927 1

n PBP NFJ 11 229614 1 12 517854 0 13 1022224 5 14 1753737 10 15 2598215 7 16 3312423 27 17 3626632 47 18 3413313 121 19 2765623 373 20 1930324 925

n PBP NFJ 21 1160977 1972 22 600372 3346 23 265865 4356 24 100565 4256 25 32250 3054 26 8688 1715 27 1917 665 28 348 182 29 50 39 30 7 6

n PBP NFJ 31 2 2

Since the maximum number of board positions at any jump is 3626632, and there can only be 31 jumps, modern computers can easily examine all game positions in a reasonable time.