Peg solitaire

The Princess of Soubise playing solitaire, 1687

Peg solitaire (or Solo Noble) is a board game for one player involving movement of pegs on a board with holes. Some sets use marbles in a board with indentations. 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).

The first evidence of the game can be traced back to the court of Louis XIV, and the specific date of 1687, with an engraving made that year by Claude Auguste Berey of Anne de Rohan-Chabot, Princess of Soubise, with the puzzle by her side. The August, 1687 edition of the French literary magazine Mercure galant contains a description of the board, rules and sample problems. This is the first known reference to the game in print.

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.

Board

English peg solitaire board
European peg solitaire board

There are two traditional boards ('.' as an initial peg, 'o' as an initial hole):

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

Play

A man playing peg solitaire

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, · indicates a peg in a hole, * emboldened indicates the peg to be moved, and o indicates an empty hole. A blue ¤ is the hole the current peg moved from; a red * is the final position of that peg, a red o is the hole of the peg that was jumped and removed.

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

* · o  →  ¤ o *  Jump to right
o · *  →  * o ¤  Jump to left
*     ¤
·  →  o  Jump down
o     *
o     *
·  →  o  Jump up
*     ¤

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

    · · ·             · · ·             · · ·             · · · 
    · * ·             · ¤ ·             · o ·             · * · 
· · · · · · ·     · · · o · · ·     · ¤ o * · · ·     · o o o · · ·
· · · o · · ·     · · · * · · ·     · · · · · · ·     · · · ¤ · · ·
· · · · · · ·     · · · · · · ·     · · · · · · ·     · · · · · · ·
    · · ·             · · ·             · · ·             · · ·
    · · ·             · · ·             · · ·             · · ·

Strategy

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.

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.

There is no solution to the European board with the initial hole centrally located, if only orthogonal moves are permitted. This is easily seen as follows, by an argument from Hans Zantema. Divide the positions of the board into A, B and C positions as follows:

    A B C
  A B C A B
A B C A B C A
B C A B C A B
C A B C A B C
  B C A B C
    A B C

Initially with only the central position free, the number of covered A positions is 12, the number of covered B positions is 12, and also the number of covered C positions is 12. After every move the number of covered A positions increases or decreases by one, and the same for the number of covered B positions and the number of covered C positions. Hence after an even number of moves all these three numbers are even, and after an odd number of moves all these three numbers are odd. Hence a final position with only one peg can not be reached: then one of these numbers is one (the position of the peg, one is odd), while the other two numbers are zero, hence even.

There are, however, several other configurations where a single initial hole can be reduced to a single peg.

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.:

* · o      ¤ o *      o * ·      * o ¤
  ·     →    ·    →     o     →    o
  ·          ·          ¤          o

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.

Studies on peg solitaire

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). They introduced a notion called pagoda function which is a strong tool to show the infeasibility of a given (generalized) peg solitaire problem. A problem for finding a pagoda function (which concludes the infeasibility of a given problem) is formulated as a linear programming problem and solvable in polynomial time (see Kiyomi and Matsui 2001). Uehara and Iwata (1990) dealt with the generalized Hi-Q problems which are equivalent to the peg solitaire problems and showed their NP-completeness. Avis and Deza (1996) formulated a peg solitaire problem as a combinatorial optimization problem and discussed the properties of the feasible region called 'a solitaire cone'. Kiyomi and Matsui (2001) proposed an efficient method for solving peg solitaire problems.

An unpublished study from 1989 on a generalised version of the game on the English board showed that each possible problem in the generalised game has 29 possible distinct solutions, excluding symmetries, as the English board contains 9 distinct 3×3 sub-squares. One consequence of this analysis is to put a lower bound on the size of possible 'inverted position' problems, in which the cells initially occupied are left empty and vice versa. Any solution to such a problem must contain a minimum of 11 moves, irrespective of the exact details of the problem.

It can be proved using abstract algebra that there are only 5 fixed board positions where the game can successfully end with one peg.[1]

Solutions to the English game

The shortest solution to the standard English game involves 18 moves, counting multiple jumps as single moves:

This solution was found in 1912 by Ernest Bergholt and proven to be the shortest possible by John Beasley in 1964.[2]

This solution can also be seen on a page that also introduces the Wolstenholme notation, which is designed to make memorizing the solution easier.

Other solutions include the following list. In these, the notation used is

In 2016, an Android game[3] was launched that included a state-space based solver for any board arrangement.

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

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 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
11 229,614 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.

It is also possible to generate all board positions. The results below have been obtained using the mcrl2 toolset (see the peg_solitaire example in the distribution).

n PBP
1 1
2 4
3 12
4 60
5 296
6 1,338
7 5,648
8 21,842
n PBP
9 77,559
10 249,690
11 717,788
12 1,834,379
13 4,138,302
14 8,171,208
15 14,020,166
16 20,773,236
n PBP
17 26,482,824
18 28,994,876
19 27,286,330
20 22,106,348
21 15,425,572
22 9,274,496
23 4,792,664
24 2,120,101
n PBP
25 800,152
26 255,544
27 68,236
28 14,727
29 2,529
30 334
31 32
32 5

Solutions to the European game

There are 3 initial non-congruent positions that have solutions. These are:

1)

          0 1 2 3 4 5 6
        0     o · ·
        1   · · · · ·
        2 · · · · · · ·
        3 · · · · · · ·
        4 · · · · · · ·
        5   · · · · ·
        6     · · ·

Possible solution: [2:2-0:2, 2:0-2:2, 1:4-1:2, 3:4-1:4, 3:2-3:4, 2:3-2:1, 5:3-3:3, 3:0-3:2, 5:1-3:1, 4:5-4:3, 5:5-5:3, 0:4-2:4, 2:1-4:1, 2:4-4:4, 5:2-5:4, 3:6-3:4, 1:1-1:3, 2:6-2:4, 0:3-2:3, 3:2-5:2, 3:4-3:2, 6:2-4:2, 3:2-5:2, 4:0-4:2, 4:3-4:1, 6:4-6:2, 6:2-4:2, 4:1-4:3, 4:3-4:5, 4:6-4:4, 5:4-3:4, 3:4-1:4, 1:5-1:3, 2:3-0:3, 0:2-0:4]

2)

          0 1 2 3 4 5 6
        0     · · ·
        1   · · o · ·
        2 · · · · · · ·
        3 · · · · · · ·
        4 · · · · · · ·
        5   · · · · ·
        6     · · ·

Possible solution: [1:1-1:3, 3:2-1:2, 3:4-3:2, 1:4-3:4, 5:3-3:3, 4:1-4:3, 2:1-4:1, 2:6-2:4, 4:4-4:2, 3:4-1:4, 3:2-3:4, 5:1-3:1, 4:6-2:6, 3:0-3:2, 4:5-2:5, 0:2-2:2, 2:6-2:4, 6:4-4:4, 3:4-5:4, 2:3-2:1, 2:0-2:2, 1:4-3:4, 5:5-5:3, 6:3-4:3, 4:3-4:1, 6:2-4:2, 3:2-5:2, 4:0-4:2, 5:2-3:2, 3:2-1:2, 1:2-1:4, 0:4-2:4, 3:4-1:4, 1:5-1:3, 0:3-2:3]

and 3)

          0 1 2 3 4 5 6
        0     · · ·
        1   · · · · ·
        2 · · · o · · ·
        3 · · · · · · ·
        4 · · · · · · ·
        5   · · · · ·
        6     · · ·

Possible solution: [2:1-2:3, 0:2-2:2, 4:1-2:1, 4:3-4:1, 2:3-4:3, 1:4-1:2, 2:1-2:3, 0:4-0:2, 4:4-4:2, 3:4-1:4, 6:3-4:3, 1:1-1:3, 4:6-4:4, 5:1-3:1, 2:6-2:4, 1:4-1:2, 0:2-2:2, 3:6-3:4, 4:3-4:1, 6:2-4:2, 2:3-2:1, 4:1-4:3, 5:5-5:3, 2:0-2:2, 2:2-4:2, 3:4-5:4, 4:3-4:1, 3:0-3:2, 6:4-4:4, 4:0-4:2, 3:2-5:2, 5:2-5:4, 5:4-3:4, 3:4-1:4, 1:5-1:3]

Board variants

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.

A common triangular variant has five pegs on a side. A solution where the final peg arrives at the initial empty hole is not possible for a hole in one of the three central positions. An empty corner-hole setup can be solved in ten moves, and an empty midside-hole setup in nine (Bell 2008):

References

  1. ↑ Mathematics and brainvita
  2. ↑ For Beasley's proof see Winning Ways, volume #4 (second edition).
  3. ↑ Peg Solitaire Solver http://www.dasmic.com/android/brainvita/index.html

Bibliography

  • Avis, D.; Deza, A. (2001), "On the solitaire cone and its relationship to multi-commodity flows", Mathematical Programming, 90 (1): 27–57, doi:10.1007/PL00011419 .
  • Beasley, John D. (1985). The Ins & Outs of Peg Solitaire. Oxford University Press. ISBN 978-0198532033. 
  • Bell, G. I. (2008), "Solving triangular peg solitaire", Journal of Integer Sequences, 11: Article 08.4.8, arXiv:math.CO/0703865Freely accessible .
  • Berlekamp, E. R.; Conway, J. H.; Guy, R. K. (1982), Winning Ways for your Mathematical Plays, London: Academic Press .
  • de Bruijn, N. G. (1972), "A solitaire game and its relation to a finite field", Journal of Recreational Mathematics, 5: 133–137 .
  • Cross, D. C. (1968), "Square solitaire and variations", Journal of Recreational Mathematics, 1: 121–123 .
  • Gardner, M., "Mathematical games", Scientific American  206 (6): 156–166, June 1962; 214 (2): 112–113, Feb. 1966; 214 (5): 127, May 1966.
  • Kiyomi, M.; Matsui, T. (2001), "Integer programming based algorithms for peg solitaire problems", Proc. 2nd Int. Conf. Computers and Games (CG 2000), Lecture Notes in Computer Science, 2063, pp. 229–240, doi:10.1007/3-540-45579-5_15 .
  • Uehara, R.; Iwata, S. (1990), "Generalized Hi-Q is NP-complete", Trans. IEICE, 73: 270–273 .
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