Peg solitaire

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Peg Solitaire is a board 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. It is also referred to as Brainvita (especially in India). Some sets use marbles in a board with indentations.

According to a popular story, the game was invented by a French aristocrat in the 17th century, when incarcerated in the Bastille. John Beasley (author of "The Ins and Outs of Peg Solitaire") has extensively searched for evidence to support this, and has found it lacking. The first reference to this story appeared in 1810, more than a hundred years after the alleged event. He believes that the colorful tale is fiction, yet it persists. In other sources, the invention of the game is attributed to the American Indians--there is also no evidence to support this.

The first evidence of the game can be traced back to the court of Louis XIV, and the specific date of 1697. Several works of art from that time show peg solitaire boards, demonstrating that the game was highly fashionable.

There are two traditional boards:

   English          European
    · · ·             · · ·
    · · ·           · · · · ·
· · · · · · ·     · · · · · · ·
· · · o · · ·     · · · o · · ·
· · · · · · ·     · · · · · · ·
    · · ·           · · · · ·
    · · ·             · · ·

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.

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.

The shortest solution to 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. This solution is shortest because it involves touching only 18 pegs (not counting those removed from the board). The solution was found in 1912 by Ernest Bergholt and proven to be the shortest possible by John Beasley in 1964. For Beasley's proof see Winning Ways, volume #4 (second edition).

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. However the reversed solution is longer than 18 moves.

There is probably no solution to the European board with the initial hole centrally located, if only orthogonal moves are permitted. [1] There are, however, several other configurations where a single initial hole can be reduced to a single peg. [2]

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 an 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 0-12-091102-7 in the UK and ISBN 1-56881-144-6 in the US (Vol 4, 2nd edition).

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.

[edit] 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

Diagrams of many solutions can be found here.

[edit] 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.

The above sequence "PBP" has been entered as A112737 in OEIS. Note that the total number of reachable board positions (sum of the sequence) is 23,475,688, while the total number of possible board positions is 2^33, or approximately 2^33/8 ~ 1 billion when symmetry is taken into account. So only about 2.2% of all possible board positions can be reached starting with the center vacant.

[edit] External links

• Peg Solitaire for Windows: Free Windows PC software that lets you play peg solitaire and view solutions.
• Blobs Peg Solitaire: Free peg solitaire game for windows with 100 prefab boards and a new board generator.
• Peg Solitaire Online Game: Beautiful Peg Solitaire game
• Peg Solitaire Solution: Easy to use solution for Peg Solitaire. Win in only 31 moves. By Kevin Paul Scarrott, Stavanger, Norway.
• Nice easy to follow .avi animation. Great for those who just want a quick visual hint of the Peg Solitaire solution (Windows Media Player format, file size 2.8MB).
• Desktop Solitaire A Peg Solitaire game for windows.
• Several pages on www.cut-the-knot.org, e.g. peg solitaire, reverse solitaire and peg solitaire and group theory
• Free Peg Solitaire Online Games Directory of Free Peg Solitaire Games
• Play Peg Solitaire Online including variants and solutions such as the English Cross.
• English Cross Puzzle: Play English Cross puzzle online (java)
• Study of Peg Solitaire [translation from French in progress]
• Solitaire and other puzzles (Freeware for mobile phones)
• Theory of the game with many solutions
• The Games and Puzzles Journal, Issue 28, September 2003