Board
There are two traditional boards ('.' as an initial peg, 'o' as an initial hole):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 (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 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 theirSolutions to the English game
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 possibility of the same pawn moved to make a further jump (No Further Jumps). NOTE: If one board position can be rotated and/or flipped into another board position, the board positions are counted as identical. Since 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 oeis:A112737, A112737 inSolutions 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):Video game
On June 26, 1992, a video game based on peg solitaire was released for the Game Boy. Titled simply "Solitaire", the game was developed by Hect. In North America, DTMC released the game as "Lazlos' Leap".In popular culture
References
Further reading
* *. * * * 206 (6): 156–166, June 1962; 214 (2): 112–113, Feb. 1966; 214 (5): 127, May 1966. *External links
* *{{cbignore *Play Multiple Versions of Peg Solitaire including English, European, Triangular, Hexagonal, Propeller, Minimum, 4Holes, 5Holes, Easy Pinwheel, Banzai7, Megaphone, Owl, Star and Arrow at