Peg solitaire
Encyclopedia
Peg solitaire is a board game
Board game
A board game is a game which involves counters or pieces being moved on a pre-marked surface or "board", according to a set of rules. Games may be based on pure strategy, chance or a mixture of the two, and usually have a goal which a player aims to achieve...
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
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
where the card games are called Patience
Solitaire
Solitaire is any tabletop game which one can play by oneself or with other people. The solitaire card game Klondike is often known as simply Solitaire....
. It is also referred to as Brainvita (especially in India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
).
According to a popular story, the game was invented by a French aristocrat in the 17th century, when incarcerated in the Bastille
Bastille
The Bastille was a fortress in Paris, known formally as the Bastille Saint-Antoine. It played an important role in the internal conflicts of France and for most of its history was used as a state prison by the kings of France. The Bastille was built in response to the English threat to the city of...
, explaining the game's less common name Solo Noble. 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 Native Americans—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, with an engraving made that year by Claude Auguste Berey of Anne de Rohan-Chabot
Anne de Rohan-Chabot
Anne de Rohan-Chabot was a French noble. A member of the House of Rohan, she was wife of the Prince of Soubise. It was she would bought the lordship of Soubise into the junior line of the Rohans. She was a short term mistress of Louis XIV...
, Princess of Soubise, with the puzzle by her side. Several works of art from that time show peg solitaire boards, demonstrating that the game was highly fashionable.
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
There are two traditional boards, graphically depicted as follows ('.' as an initial peg, 'o' as an initial hole): |
English European
· · · · · ·
· · · · · · · ·
· · · · · · · · · · · · · ·
· · · o · · · · · · o · · ·
· · · · · · · · · · · · · ·
· · · · · · · ·
· · · · · ·
Play
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 green is the hole the current peg moved from; a red is the final position of that peg, a red is the hole of the peg that was jumped and removed.
Thus valid moves in each of the four orthogonal directions are:
* * o → Jump to right
o * * → Jump to left
*
* → Jump down
o
o
* → Jump up
*
On an English board, the first three moves might be:
* * * * * * * * * * * *
* * * * * * 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
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 WaysWinning Ways for your Mathematical Plays
Winning Ways for your Mathematical Plays by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games...
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 the 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 2^9 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.
Solutions to the English game
The shortest solution to the standard English game involves 18 moves, counting multiple jumps as single moves:| Shortest solution to English peg solitaire |
|---|
e-x l-j c-k * * * * * * * * * * * * * * * * * o * * o * * * * * * * * * * * * * * * * * * * * * o * * * * o * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * P-f D-P G-I J-H * * o * * o * * o * * o * o * o * * o * * o * * * * * o * * * * * o o * * * * * o o * * * * * o o * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * o * * * * * * * o * * o * * * * * * * * * * * * m-G-I i-k g-i L-J-H-l-j-h * * o * * o * * o * * o * o * * o * * o * * o * * * * * o o * * o o o * o o o o * * * * * * * * * * * * o * * * * * * o * * * * * o * * * o * * * o * o o * * * o * o o * o * * o * * o * * o * * o * * * * * * * * * * * * C-K p-F A-C-K M-g-i * * o * * o * * o * * o * o * * o * * o * * o * o * o o o o o o * o o o o o o * o o o o o o o o o * * * * * o o * * * * o o * * o * * o o * o * * o o * o o o o o * o o o o o * o o o o o o * o o o o * o * o * o o * o * * o * * o o o a-c-k-I d-p-F-D-P-p o-x o o o o o o * o o o o o o o o * o o o o o o o o o o o o o o o o o * o * o o o * o o o o o o o o * o o o o o o o o o o o o o o o o * o o o o o o o o o o o o o The order of some of the moves can be exchanged. Note that if you instead think of * as a hole and o as a peg, you can solve the puzzle by following the solution in reverse, starting from the last picture, going towards the first. However, this requires more than 18 moves. |
This solution was found in 1912 by Ernest Bergholt and proven to be the shortest possible by John Beasley in 1964.
This solution can also be see 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
- 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.
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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
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...
. 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).
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Solutions to the European game
There are 3 initial non-congruentCongruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
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):
| Shortest solution to triangular variant |
|---|
* = peg to move next; = hole created by move; = jumped peg removed; = hole filled by jumping; * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * o * * * * o * o * o * o * * o * o o o o o * * o o o o o o o o o o o * * o o o o o o * o o o o * * o * o o * o o o o * o o o o o o o o |
Literature
External links
- Peg solitaire and group theory
- Peg solitaire solver which determines all possible solutions for arbitrary solitaire boards
- Interactive Google App Engine application: discover the solution by yourself
- The Games and Puzzles Journal, Issue 28, September 2003
- Javascript/HTML based animation of solution to peg solitaire.
- Peg solitaire, step by step solution by Kevin Paul Scarrott
- Firsts and Lasts of Peg Solitaire
- Various types of Peg Solitaire game online