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==Strategy== There are many different solutions to the standard problem, and one notation used to describe them assigns letters to the holes (although numbers may also be used): 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.: '''*''' ยท o {{blue|ยค}} {{red|'''o'''}} {{red|'''*'''}} o {{red|'''*'''}} ยท {{red|'''*'''}} {{red|'''o'''}} {{blue|ยค}} ยท โ ยท โ {{red|'''o'''}} โ o ยท ยท {{blue|ยค}} o This technique can be used with a line of 3, a block of 2ยท3 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.<ref name="conway">{{citation|last1=Berlekamp |first1=E. R. |author1-link=Elwyn R. Berlekamp |last2=Conway |first2=J. H. |author2-link=John H. Conway |last3=Guy |first3=R. K. |author3-link=Richard K. Guy |title=Winning Ways for your Mathematical Plays |date=2001 |orig-year=1981 |publisher=A K Peters/CRC Press |isbn=978-1568811307 |edition=2nd |oclc=316054929|language=en |title-link=Winning Ways for your Mathematical Plays }}</ref> 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.<ref name="kiyomi">{{citation |last1=Kiyomi |first1=M. |last2=Matsui |first2=T. |title=Proc. 2nd Int. Conf. Computers and Games (CG 2000): Integer programming based algorithms for peg solitaire problems |series=Lecture Notes in Computer Science |volume=2063 |year=2001 |pages=229โ240 |doi=10.1007/3-540-45579-5_15 |chapter=Integer Programming Based Algorithms for Peg Solitaire Problems |isbn=978-3-540-43080-3 |citeseerx=10.1.1.65.6244 }}</ref> A paper in 1990 dealt with the generalized Hi-Q problems which are equivalent to the peg solitaire problems and showed their [[NP-completeness]].<ref name="uehara">{{cite journal|last1=Uehara|first1=R.|last2=Iwata|first2=S.|title=Generalized Hi-Q is NP-complete|journal=Trans. IEICE|volume=73|year=1990|pages=270โ273}}</ref> 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'.<ref name="avis">{{citation |last1=Avis |first1=D. |author1-link=David Avis |last2=Deza |first2=A. |title=On the solitaire cone and its relationship to multi-commodity flows |journal=Mathematical Programming |volume=90 |issue=1 |year=2001 |pages=27โ57 |doi=10.1007/PL00011419|s2cid=7852133 }}</ref> 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.<ref name="spielverderber">{{citation |last1=Eichler |last2=Jรคger |last3=Ludwig |title=c't 07/1999 Spielverderber, Solitaire mit dem Computer lรถsen |language=de |volume=7 |year=1999 |pages=218<!--page of the reference or pages in the book?-->}}</ref> In 2001 an efficient method for solving peg solitaire problems was developed.<ref name="kiyomi"/> 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 2<sup>9</sup> possible distinct solutions, excluding symmetries, as the English board contains 9 distinct 3ร3 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.<ref>{{citation| title=Mathematics and brainvita | website=Notes on Mathematics | date=28 August 2012 | url=https://notesonmathematics.wordpress.com/2012/08/28/the-mathematics-of-brainvita/ | access-date=6 September 2018}}</ref> === Solutions to the English game === <imagemap> File:Peg_Solitaire_interactive_solution_guide.svg|thumb|Interactive solution guide for English Peg Solitaire. default [https://upload.wikimedia.org/wikipedia/commons/9/99/Peg_Solitaire_interactive_solution_guide.svg] </imagemap> The shortest solution to the standard English game involves 18 moves, counting multiple jumps as single moves: {| class="wikitable collapsible collapsed" ! Shortest solution to English peg solitaire |- | e-x l-j c-k ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท {{blue|ยค}} ยท '''*''' ยท ยท {{blue|ยค}} ยท ยท o ยท ยท o {{red|'''o'''}} ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท {{red|'''o'''}} ยท ยท ยท ยท ยท ยท {{red|'''*'''}} {{red|'''o'''}} {{blue|ยค}} ยท ยท ยท ยท ยท {{red|'''*'''}} o ยท ยท ยท ยท o ยท ยท ยท ยท ยท ยท {{red|'''*'''}} ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท P-f D-P G-I J-H ยท ยท o ยท ยท o ยท ยท o ยท ยท o ยท o {{red|'''*'''}} ยท o ยท ยท o ยท ยท o ยท ยท ยท ยท ยท {{red|'''o'''}} o ยท ยท ยท ยท ยท o o ยท ยท ยท ยท ยท o o ยท ยท ยท ยท ยท o o ยท ยท ยท ยท ยท {{blue|ยค}} ยท ยท ยท ยท ยท ยท {{red|'''*'''}} ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท {{red|'''o'''}} ยท ยท ยท ยท ยท ยท {{red|'''*'''}} {{red|'''o'''}} {{blue|ยค}} ยท ยท ยท {{blue|ยค}} {{red|'''o'''}} {{red|'''*'''}} o ยท ยท ยท ยท ยท {{blue|ยค}} ยท ยท o ยท ยท o ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท m-G-I i-k g-i L-J-H-l-j-h ยท ยท o ยท ยท o ยท ยท o ยท ยท o ยท o ยท ยท o ยท ยท o ยท ยท o ยท ยท ยท ยท ยท o o {{blue|ยค}} ยท ยท {{blue|ยค}} {{red|'''o'''}} {{red|'''*'''}} o o {{blue|ยค}} {{red|'''o'''}} {{red|'''*'''}} o ยท o o o {{red|'''*'''}} {{red|'''o'''}} {{blue|o}} {{red|'''o'''}} {{blue|o}} o ยท ยท ยท ยท ยท ยท {{red|'''o'''}} ยท ยท ยท ยท ยท ยท o ยท ยท ยท ยท ยท ยท o ยท ยท ยท ยท ยท {{red|'''o'''}} o ยท ยท ยท o {{red|'''*'''}} {{red|'''o'''}} {{blue|o}} ยท ยท ยท o ยท o o ยท ยท ยท o ยท o o ยท {{blue|ยค}} {{red|'''o'''}} {{blue|o}} {{red|'''o'''}} {{blue|o}} o ยท ยท o ยท ยท o ยท ยท o ยท ยท o ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท C-K p-F A-C-K M-g-i ยท ยท o ยท ยท o ยท ยท o ยท ยท o ยท o ยท ยท o ยท ยท o ยท ยท o ยท o ยท o o o o o o ยท o o o o o o ยท o o o o o {{blue|o}} {{red|'''o'''}} {{red|'''*'''}} o o o o ยท ยท ยท ยท ยท o o ยท ยท {{blue|ยค}} ยท ยท o o ยท ยท o ยท ยท o o {{red|'''o'''}} ยท o ยท ยท o o ยท o {{red|'''*'''}} o o o o ยท o {{red|'''o'''}} o o o o ยท o {{red|'''*'''}} o o o o {{blue|ยค}} o ยท o o o o {{red|'''o'''}} ยท o {{red|'''*'''}} ยท o {{red|'''o'''}} ยท o o ยท o {{blue|ยค}} ยท ยท o ยท ยท {{blue|o}} {{red|'''o'''}} {{blue|ยค}} o o o a-c-k-I d-p-F-D-P-p o-x {{blue|ยค}} {{red|'''o'''}} {{blue|o}} o o o o o o ยท o {{red|'''o'''}} {{blue|ยค}} o o o o o o o ยท o {{blue|o}} o o o o {{red|'''o'''}} o o o o o o o o o o o o ยท o ยท {{red|'''o'''}} o o o ยท {{red|'''*'''}} {{red|'''o'''}} {{blue|o}} o o o {{blue|ยค}} {{red|'''o'''}} {{red|'''*'''}} o o o o o ยท o {{red|'''*'''}} o o o o {{red|'''o'''}} o {{red|'''o'''}} o o o o o o o o o o ยท o {{blue|o}} {{red|'''o'''}} {{blue|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.<ref>For Beasley's proof see ''[[Winning Ways for your Mathematical Plays|Winning Ways]],'' volume #4 (second edition).</ref> This solution can also be seen on [https://web.archive.org/web/20140520101224/http://www.topaccolades.com/notation/solitaire.htm 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 (the pegs jumped over are left as an exercise to the reader) *, or / (''a slash is used to separate 'chunks' such as a six-purge out'') <pre> 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 </pre> [[File:easy_solitaire_solution.svg|thumb|upright=1.5|An easily remembered solution of first clearing edges by focusing on the holes circled in white – in figure 1, pegs are labelled in the order they are removed]] ===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 ('''P'''ossible '''B'''oard '''P'''ositions) of possible board positions after '''n''' jumps, and the possibility of the same peg moved to make a further jump ('''N'''o '''F'''urther '''J'''umps). Interesting to note is that the shortest way to fail the game is in six moves, and the solution (besides its rotations and reflections) is unique. An example of this is as follows: 4 โ 16; 23 โ 9; 14 โ 16; 17 โ 15; 19 โ 17; 31 โ 23. (In this notation, the pegs are numbered from left to right, starting with 0, and moving down each row and to the far left once each row is marked.) '''NOTE: If one board position can be rotated and/or flipped into another board position, the board positions are counted as identical.''' {{col-begin|width=auto}} {{col-break}} {| class="wikitable" |- ! 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 |} {{col-break|gap=2em}} {| class="wikitable" |- ! 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 |} {{col-break|gap=2em}} {| class="wikitable" |- ! 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 |} {{col-break|gap=2em}} {| class="wikitable" |- ! n || PBP || NFJ |- | 31 || 2 || 2 |} {{col-end}} Since there can only be 31 jumps, modern computers can easily examine all game positions in a reasonable time.<ref name="solboard">{{cite web|author=<!--Not stated-->|date=2020-08-31|title=solboard|url=https://github.com/vsaugey/solboard|access-date=2020-08-31|website=github|quote=Implementation of brute force calculation of the Peg solitaire game}}</ref> The above sequence "PBP" has been entered as [[oeis:A112737|A112737]] in [[On-Line Encyclopedia of Integer Sequences|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 8,589,934,590 (33bit-1) (2^33), 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). {{col-begin|width=auto}} {{col-break}} {| class="wikitable" |- ! n || PBP |- | 1 || 1 |- | 2 || 4 |- | 3 || 12 |- | 4 || 60 |- | 5 || 296 |- | 6 || 1,338 |- | 7 || 5,648 |- | 8 || 21,842 |} {{col-break|gap=2em}} {| class="wikitable" |- ! 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 |} {{col-break|gap=2em}} {| class="wikitable" |- ! 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 |} {{col-break|gap=2em}} {| class="wikitable" |- ! n || PBP |- | 25 || 800,152 |- | 26 || 255,544 |- | 27 || 68,236 |- | 28 || 14,727 |- | 29 || 2,529 |- | 30 || 334 |- | 31 || 32 |- | 32 || 5 |} {{col-end}} In the results below, it has generated all the board positions it '''really''' reached starting with the center vacant and finishing in the central hole. {{col-begin|width=auto}} {{col-break}} {| class="wikitable" |- ! n || Real |- | 1 || 1 |- | 2 || 4 |- | 3 || 12 |- | 4 || 60 |- | 5 || 292 |- | 6 || 1,292 |- | 7 || 5,012 |- | 8 || 16,628 |} {{col-break|gap=2em}} {| class="wikitable" |- ! n || Real |- | 9 || 49,236 |- | 10 || 127,964 |- | 11 || 285,740 |- | 12 || 546,308 |- | 13 || 902,056 |- | 14 || 1,298,248 |- | 15 || 1,639,652 |- | 16 || 1,841,556 |} {{col-break|gap=2em}} {| class="wikitable" |- ! n || Real |- | 17 || 1,841,556 |- | 18 || 1,639,652 |- | 19 || 1,298,248 |- | 20 || 902,056 |- | 21 || 546,308 |- | 22 || 285,740 |- | 23 || 127,964 |- | 24 || 49,236 |} {{col-break|gap=2em}} {| class="wikitable" |- ! n || Real |- | 25 || 16,628 |- | 26 || 5,012 |- | 27 || 1,292 |- | 28 || 292 |- | 29 || 60 |- | 30 || 12 |- | 31 || 4 |- | 32 || 1 |} {{col-end}} ===Solutions to the European game=== There are 3 initial [[Congruence (geometry)|non-congruent]] positions that have solutions.<ref>{{citation |language=fr |last=Brassine |first=Michel |date=December 1981 |title=Dรฉcouvrez... le solitaire |magazine=Jeux et Stratรฉgie}}</ref> These are: 1) 0 1 2 3 4 5 6 0 o ยท ยท 1 ยท ยท ยท ยท ยท 2 ยท ยท ยท ยท ยท ยท ยท 3 ยท ยท ยท ยท ยท ยท ยท 4 ยท ยท ยท ยท ยท ยท ยท 5 ยท ยท ยท ยท ยท 6 ยท ยท ยท 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 ยท ยท ยท 1 ยท ยท o ยท ยท 2 ยท ยท ยท ยท ยท ยท ยท 3 ยท ยท ยท ยท ยท ยท ยท 4 ยท ยท ยท ยท ยท ยท ยท 5 ยท ยท ยท ยท ยท 6 ยท ยท ยท 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 ยท ยท ยท 1 ยท ยท ยท ยท ยท 2 ยท ยท ยท o ยท ยท ยท 3 ยท ยท ยท ยท ยท ยท ยท 4 ยท ยท ยท ยท ยท ยท ยท 5 ยท ยท ยท ยท ยท 6 ยท ยท ยท 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. In 2025, Michael Stevens laid out every variant on one board. He called it ''Omnijump'', and he put it in the [https://www.curiositybox.com/?utm_term=&utm_source=google&utm_medium=cpc&utm_campaign=CB_US_Search_Brand&utm_content=US_Brand_Exact_Match&isbrand=true&gad_source=1&gclid=Cj0KCQjwkN--BhDkARIsAD_mnIotT4arHu0DPi2TtqH5t7RsLlNsFjsG4DAQmmZVwYHWRTQ5T5Yb1OMaArlHEALw_wcB/ 2025 Spring Curiosity box].[https://www.youtube.com/shorts/HdX6dNIlCQI] [[Image:Peg Solitaire game board shapes.svg|frame|none|Peg solitaire game board shapes:<br/> (1) French (European) style, 37 holes, 17th century;<br/> (2) J. C. Wiegleb, 1779, Germany, 45 holes;<br/> (3) Asymmetrical 3-3-2-2 as described by George Bell, 20th century;<ref>See ''Generalized Cross Boards'' in: [http://www.gibell.net/pegsolitaire/#gencross George's Peg Solitaire Page]</ref><br/> (4) English style (standard), 33 holes;<br/> (5) Diamond, 41 holes;<br/> (6) Triangular, 15 holes.<br/> Grey = the hole for the survivor.]] 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): {| class="wikitable collapsible collapsed" ! Shortest solution to triangular variant |- | '''*''' = peg to move next; {{blue|ยค}} = hole created by move; {{red|'''o'''}} = jumped peg removed; {{blue|'''*'''}} = hole filled by jumping; ยท ยท ยท '''*''' {{blue|ยค}} ยท ยท ยท ยท ยท ยท ยท '''*''' {{red|'''o'''}} {{blue|ยค}} ยท ยท ยท ยท ยท ยท '''*''' ยท ยท {{blue|ยค}} ยท ยท {{blue|'''*'''}} {{red|'''o'''}} ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท ยท {{red|'''o'''}} ยท ยท '''*''' {{blue|'''*'''}} ยท '''*''' '''*''' ยท o ยท ยท {{blue|ยค}} {{red|'''o'''}} {{blue|'''*'''}} '''*''' ยท o {{blue|'''*'''}} {{red|'''o'''}} {{blue|ยค}} ยท o ยท {{blue|'''*'''}} o ยท o ยท ยท o ยท o o o o o {{blue|'''*'''}} {{blue|'''*'''}} '''*''' '''*''' {{blue|ยค}} {{blue|ยค}} o o o o {{red|'''o'''}} o {{red|'''o'''}} o {{blue|'''*'''}} {{blue|'''*'''}} o {{red|'''o'''}} {{red|'''o'''}} o o {{blue|'''*'''}} o o {{blue|ยค}} {{blue|ยค}} ยท ยท {{blue|ยค}} o {{red|'''o'''}} {{red|'''o'''}} o o o {{blue|'''*'''}} {{blue|'''*'''}} o o ยท {{red|'''o'''}} o o {{red|'''o'''}} o o '''*''' '''*''' o ยท o {{blue|ยค}} {{blue|ยค}} o ยท o o o o '''*''' o o o o {{blue|ยค}} o o {{blue|'''*'''}} o o |}
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