Peg solitaire

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Image:Solitaire 01.jpg Image:Solitaire.JPG 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. Some sets use marbles in a board with indentations.

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

The game really took off when Queen Victoria started enjoying it in England.

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: Solution details follow.

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:

e-x
l-j
c-k
P-f
D-P
G-I
J-H
m-G-I
i-k
g-i
L-J-H-l-j-h
C-K
p-F
A-C-K
M-g-i
a-c-k-I
d-p-F-D-P-p
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 provably 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 01120911027 in the UK and ISBN 1568811446 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.

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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.

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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.

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