Peg solitaire

Peg Solitaire, Solo Noble, Solo Goli, Marble Solitaire or simply Solitaire 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 as solitaire in Britain and as peg solitaire in the US where 'solitaire' is the common name for patience.
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 ten years later by Claude Auguste Berey of Anne de Rohan-Chabot, Princess of Soubise, with the puzzle by her side. The August 1697 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

There are two traditional boards :

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 blue 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

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 cannot be reached, since that would require that one of these numbers is one, 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 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 known. This analysis introduced a notion called pagoda function which is a strong tool to show the infeasibility of a given generalized peg solitaire problem.
A solution for finding a pagoda function, which demonstrates the infeasibility of a given problem, is formulated as a linear programming problem and solvable in polynomial time.
A paper in 1990 dealt with the generalized Hi-Q problems which are equivalent to the peg solitaire problems and showed their NP-completeness.
A 1996 paper formulated a peg solitaire problem as a combinatorial optimization problem and discussed the properties of the feasible region called 'a solitaire cone'.
In 1999 peg solitaire was completely solved on a computer using an exhaustive search through all possible variants. It was achieved making use of the symmetries, efficient storage of board constellations and hashing.
In 2001 an efficient method for solving peg solitaire problems was developed.
An unpublished study from 1989 on a generalized version of the game on the English board showed that each possible problem in the generalized 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.

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 seen on , 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
, or /

 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