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 simply as Solitaire in the United Kingdom 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.

Template:Solution 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

To see the above solution illustrated, go to Solution to peg solitaire.

Note that following the solution in reverse, that is, o-x, P-p, D-P, F-D, p-F, d-p, k-I, etc... also works.

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 starting 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 three permit. Thus a hole at a can only leave a single peg at a, p, O 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 triangular 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.

Some solutions

In these, the notation used is

• List of starting holes
• Colon
• List of end target pegs
• Equals sign
• Source peg and destination hole (you have to work out what it jumps over yourself)
• , or / (a slash is used to separate 'chunks' such as a six-purge out)

x:x=ex,lj,ck,Pf,DP,GI,JH,mG,GI,ik,gi,LJ,JH,Hl,lj,jh,CK,pF,AC,CK,Mg,gi,ac,ck,kI,dp,pF,FD,DP,Pp,ox
x:x=ex,lj,xe/hj,Ki,jh/ai,ca,fd,hj,ai,jh/MK,gM,hL,Fp,MK,pF/CK,DF,AC,JL,CK,LJ/PD,GI,mG,JH,GI,DP/Ox
j:j=lj,Ik,jl/hj,Ki,jh/mk,Gm,Hl,fP,mk,Pf/ai,ca,fd,hj,ai,jh/MK,gM,hL,Fp,MK,pF/CK,DF,AC,JL,CK,LJ/Jj
i:i=ki,Jj,ik/lj,Ik,jl/AI,FD,CA,HJ,AI,JH/mk,Hl,Gm,fP,mk,Pf/ai,ca,fd,hj,ai,jh/gi,Mg,Lh,pd,gi,dp/Ki
e:e=xe/lj,Ik,jl/ck,ac,df,lj,ck,jl/GI,lH,mG,DP,GI,PD/AI,FD,CA,JH,AI,HJ/pF,MK,gM,JL,MK,Fp/hj,ox,xe
d:d=fd,xe,df/lj,ck,ac,Pf,ck,jl/DP,KI,PD/GI,lH,mG,DP,GI,PD/CK,DF,AC,LJ,CK,JL/MK,gM,hL,pF,MK,Fp/pd
b:b=jb,lj/ck,ac,Pf,ck/DP,GI,mG,JH,GI,PD/LJ,CK,JL/MK,gM,hL,pF,MK,Fp/xo,dp,ox/xe/AI/BJ,JH,Hl,lj,jb
b:x=jb,lj/ck,ac,Pf,ck/DP,GI,mG,JH,GI,PD/LJ,CK,JL/MK,gM,hL,pF,MK,Fp/xo,dp,ox/xe/AI/BJ,JH,Hl,lj,ex
a:a=ca,jb,ac/lj,ck,jl/Ik,pP,KI,lj,Ik,jl/GI,lH,mG,DP,GI,PD/CK,DF,AC,LJ,CK,JL/dp,gi,pd,Mg,Lh,gi/ia
a:p=ca,jb,ac/lj,ck,jl/Ik,pP,KI,lj,Ik,jl/GI,lH,mG,DP,GI,PD/CK,DF,AC,LJ,CK,JL/dp,gi,pd,Mg,Lh,gi/dp

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 2,757 0 8 9,751 0 9 31,312 0 10 89,927 1

n PBP NFJ 11 22,9614 1 12 517,854 0 13 1,022,224 5 14 1,753,737 10 15 2,598,215 7 16 3,312,423 27 17 3,626,632 47 18 3,413,313 121 19 2,765,623 373 20 1,930,324 925

n PBP NFJ 21 1,160,977 1,972 22 600,372 3,346 23 265,865 4,356 24 100,565 4,256 25 32,250 3,054 26 8,688 1,715 27 1,917 665 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 3,626,632, and there can only be 31 jumps, modern computers can easily examine all game positions in a reasonable time.

External Links

• Peg solitaire (http://www.cut-the-knot.org/proofs/pegsolitaire.shtml)
• Reverse Solitaire (http://www.cut-the-knot.org/proofs/rsolitaire.shtml)
• Peg Solitaire and Group Theory (http://www.cut-the-knot.org/proofs/PegsAndGroups.shtml)
• Free Peg Solitaire Online Games (http://freewebarcade.com/board/peg/index.html) Directory of Free Peg Solitaire Games
• Play Peg Solitaire Online (http://www.farfarfar.com/games/peg_solitaire/) including variants and solutions such as the English Cross.
• English Cross Puzzle (http://www.cheesygames.com/solcross/): Play English Cross puzzle online (java)de:Solitär (Brettspiel)

he:מחשבת ja:ペグ・ソリテール