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656IEICE TRANS. FUNDAMENTALS, VOL.E83A, NO.4 APRIL 2000

PAPER Special Section on Discrete Mathematics and Its Applications

A Combinatorial Approach to the Solitaire Game

David AVIS, Antoine DEZA, and Shmuel ONN, Nonmembers

SUMMARY The classical game of peg solitaire has uncer-tain origins, but was certainly popular by the time of Louis XIV,and was described by Leibniz in 1710. One of the classical prob-lems concerning peg solitaire is the feasibility issue. An earlytool used to show the infeasibility of various peg games is therule-of-three [Suremain de Missery 1841]. In the 1960s the de-scription of the solitaire cone [Boardman and Conway] providesnecessary conditions: valid inequalities over this cone, known aspagoda functions, were used to show the infeasibility of variouspeg games. In this paper, we recall these necessary conditionsand present new developments: the lattice criterion, which gen-eralizes the rule-of-three; and results on the strongest pagodafunctions, the facets of the solitaire cone.key words: solitaire peg game, feasibility, combinatorial ap-proach

1. Introduction and Basic Denitions

1.1 Introduction

Peg solitaire is a peg game for one player which is playedon a board containing a number of holes. The mostcommon modern version uses a cross shaped board with33 holessee Fig. 1although a 37 hole board is com-mon in France. Computer versions of the game nowfeature a wide variety of shapes, including rectanglesand triangles. Initially the central hole is empty, theothers contain pegs. If in some row (column respec-tively) two consecutive pegs are adjacent to an emptyhole in the same row (column respectively), we maymake a move by removing the two pegs and placingone peg in the empty hole. The objective of the gameis to make moves until only one peg remains in the cen-tral hole. Variations of the original game, in addition tobeing played on dierent boards, also consider variousalternate starting and nishing congurations.

The game itself has uncertain origins, and dierentlegends attest to its discovery by various cultures. Anauthoritative account with a long annotated bibliogra-phy can be found in the comprehensive book of Beasley[3]. The book mentions an engraving of Berey, dated1697, of a lady with a solitaire board. The book also

Manuscript received July 31, 1999.The author is with the School of Computer Science,

McGill University, Montreal, Quebec, H3A 2A7, Canada.The author is with the Department of Mathematical

and Computing Sciences, Tokyo Institute of Technology, To-kyo, 152-8552 Japan.

The author is with the Davidson Faculty of IE & M,Technion-Israel Institute of Technology, 32000 Haifa, Israel.

contains a quotation of Leibniz [6] which was written forthe Berlin Academy in 1710. Apparently the rst theo-retical study of the game that was published was donein 1841 by Suremain de Missery, and was reported in apaper by Vallot [8]. The modern mathematical studyof the game dates to the 1960s at Cambridge Univer-sity. The group was led by Conway who has written achapter in [4] on various mathematical aspects of thesubject. One of the problems studied by the Cambridgegroup is the following basic feasibility problem (see Def-inition 1 in the sequel for a formal denition):

Peg solitaire feasibility problem: Given a board B and apair of congurations (c, c) on B, determine if the pair(c, c) is feasible, that is, if there is a legal sequence ofmoves transforming c into c.

The complexity of the feasibility problem for thegame played on a n by n board was shown by Ueharaand Iwata [7] to be NP-complete, so easily checked nec-essary and sucient conditions for feasibility are un-likely to exist. In this paper, we recall constructionsused to prove the infeasibility of some pair (c, c): therule-of-three in Sect. 2, the solitaire cone in Sect. 3.1;and present new developments in Sects. 3.2 and 4.

1.2 Basic Denitions

In this subsection we introduce some terminology usedthroughout this paper. The board of a peg solitairegame is a nite subset B Z2. Thus, B stands forthe set of locations (i, j) of holes of the board on whichthe game is played. For example, the classical 33-boardis: B = {(i, j): 1 i 1, 3 j 3} {(i, j) :3 i 3, 1 j 1}. A conguration c on theboard is an integer vector c ZB RB . It can beinterpreted as a conguration of pegs on the board: inthe usual game, all congurations c lie in {0, 1}B, withthe interpretation that hole (i, j) B contains a pegif ci,j = 1 and is empty if ci,j = 0; extending this,we allow any integer (possibly negative) number ci,j ofpegs to occupy any hole (i, j) B. The complement ofa {0, 1}-conguration c {0, 1}B is dened to be theconguration c := l c where l = (1, 1, . . . , 1) RBis the all-ones conguration. A move or a jump is avector in RB which has 3 non-zero entries: two entriesof 1 in the positions from which pegs are removedand one entry of 1 for the hole receiving the new peg.

AVIS et al.: A COMBINATORIAL APPROACH TO THE SOLITAIRE GAME657

Fig. 1 A feasible English solitaire peg game with possible rst and last moves.

Fig. 2 The score of a nal conguration with only one peg remaining.

We can now make the peg solitaire feasibility problemprecise.

Denition 1.1: Given a board B and an associatedset of movesM, a pair (c, c) of congurations is feasibleif there is a sequence 1, . . . , k M of moves on Bsuch that

c c =k

i=1

i and c+i

j=1

j {0, 1}B for i = 1, . . . , k

For instance, the English 33-board admits 76 moves(none over the 8 corners, 24 moves over the 12 holesnext to a corner and 52 moves over the 13 remainingholes); see Fig. 1 for possible rst and last moves insome sequence of moves transforming the initial cong-uration c0 to its complementary c0.

2. The Rule-of-Three

In this section we recall the so-called rule-of-three(cf. [3], [4]), a classical construction used to test soli-taire game feasibility. The rule-of-three can be used,for example, to show that on the cross shaped En-glish 33-board, starting with the initial congurationc0 of Fig. 1, the only reachable nal congurationswith exactly one peg are c0 (given in Fig. 1), c

1, c

2,

c3 and c4 with, respectively, a nal peg in position

(0, 0), (3, 0), (0, 3), (3, 0) and (0,3).Let Z2 := {a, b, c, e} be the Abelian group with

identity e and addition table a + a = b + b = c + c =e, a+ b = c, a+ c = b, b+ c = a. Dene the followingtwo maps g1, g2 : Z2 Z2, which simply color theinteger lattice Z2 by diagonals of a, b and c in eitherdirection; see Fig. 2:

g1(i, j) :=

a if (i+ j) 0 (mod 3)b if (i+ j) 1 (mod 3)c if (i+ j) 2 (mod 3)

,

g2(i, j) :=

a if (i j) 0 (mod 3)b if (i j) 1 (mod 3)c if (i j) 2 (mod 3)

.

For each (i, j) B Z2 let ei,j be the (i, j)th unitvector in RB, and dene the score map to be the Z-module homomorphism : ZB Z22 with (ei,j) :=(g1(i, j) , g2(i, j)

). Thus, the score of a conguration

c ZB is given by

(c) =

(i,j)Bci,j

(g1(i, j) , g2(i, j)

).

Since the board B under discussion will always be clearfrom the context, we use the notation for any board.For instance, the score of the conguration c0 of one pegin the center of the English 33-board is (c0) = (a, a),as is also the score of its complement c0; see Fig. 2. Thescore of the board B (all holes lled) is dened to be(B) = (l). It is easy to verify that any feasible move on any board B has the identity score () = (e, e).This gives the following proposition.

Proposition 2.1 (The rule-of-three):A necessary condition for a pair of conguration (c, c)to be feasible is that (c c) = (e, e), namely, c c Ker().

Using Prop. 2.1, we can show that, besides the con-guration c0 given in Fig. 1, the only nal congurationc with exactly one non-zero entry ci,j = 1 forming afeasible pair (c0, c) are the 4 congurations c1, c

2, c

3

and c4. Figure 2 shows that (c) = (a, a) = (c0) if c


is one of c0, c1 . . . c4, whereas (c) = (a, a) otherwise.With c = l c the complement of c, the rule-of-

three implies that (c, c) is feasible only if (c) = (c),which is equivalent to (B) = (c) + (c) = (c) +(c) = (e, e). In other words, a necessary condition forthe congurations pair (c, c) to be feasible is that theboard score is (B) = (e, e). Such a board is called anull-class board in [3]. For example, the score of theEnglish 33-board is (B) = (c0) + (c0) = (a, a) +(a, a) = (e, e).

3. Solitaire Cone and Pagoda Functions

3.1 Solitaire Cone

A rst relaxation of the feasibility problem is to allowany integer (positive or negative) number of pegs tooccupy any hole for any intermediate congurations.We call this game the integer game, and call the originalgame the 0-1 game. Note that in a 0-1 game we requirethat for each intermediate conguration of the game ahole is either empty or contains a single peg. Clearly,c, c is integer feasible if and only if

c c ICB =M

: N

where the integer solitaire cone ICB is the set of allnon-negative integer linear combinations of moves. Un-fortunately deciding if cc can be expressed as the sumof move seems to be a hard computational problem. Weget the following necessary criterion:

Proposition 3.1 (The integer cone criterion):A necessary condition for a pair of congurations (c, c)to be feasible is that c c ICB.

A further relaxation of the game leads to a moretractable condition. In the fractional game we allowany fractional (positive or negative) number of pegs tooccupy any hole for any intermediate congurations.A fractional move is obtained by multiplying a moveby any positive scalar and is dened to correspond tothe process of adding a move to a given conguration.For example, let c = [111], c = [101]. Then c c =[010] = 12 [111]+ 12 [111] is a feasible fractionalgame and can be expressed as the sum of two fractionalmoves, but is not feasible as a 0-1 or integer game.Clearly, c, c is fractional feasible if and only if

c c CB =M

: R+

where the solitaire cone CB is the set of all non-negativelinear combinations of moves. We get the weaker, butuseful, following necessary criterion:

Fig. 3 An infeasible classical solitaire peg game.

Proposition 3.2 (The cone criterion):A necessary condition for a pair of congurations (c, c)to be feasible is that c c CB.

The condition c c CB is therefore a necessarycondition for the feasibility of the original peg gameand, more usefully, provides a certicate for the infea-sibility of certain games. The certicate of infeasibil-ity is any inequality valid for CB which is violated byc c. According to [3], page 71, these inequalitieswere developed by J.H. Conway and J.M. Boardmanin 1961, and were called pagoda functions by Conway. . . . They are also known as resource counts, and arediscussed in some detail in Conway [4]. The strongestsuch inequalities are induced by the facets of CB. Forexample, the facet (iii) (given by Beasley) of Fig. 4 in-duces an inequality a x 0 that is violated by c cwith (c, c) given in Fig. 3: (c c) a = 2 > 0. This im-plies that this game is not feasible even as a fractionalgame and, therefore, not feasible as an integer game orclassical 0-1 game either.

3.2 Facets of the Solitaire Cones

Most of the results can be applied to boards which aresubsets of the square lattice in the plane, such as theoriginal peg solitaire board. For simplicity let us con-sider rectangular boards. For n 4 or m 4, the soli-taire cone CBm,n associated to the m by n board Bm,nis a pointed full-dimensional cone and the moves of thesolitaire cone are extreme rays; see [1] for a detailedstudy of CBm,n . Let us assume that m 4 or n 4and that the holes of Bm,n are ordered in some way. Itis convenient to display cc and asm by n matrices,although of course all products should be interpreted asdot products of the corresponding mn-vectors. We rep-resent the coecients of the facet inducing inequalityaz 0 by them by n array a = [ai,j ]. It is a convenientabuse of terminology to refer to a as a facet of CB. Acorner of a is a coecient ai,j with i {1,m} andj {1, n}. The notation T = (t1, t2, t3) refer to a con-secutive triple of row or column indices. Each consecu-tive triple denes a triangle inequality : at1 at2 + at3and a triangle inequality is tight if equality holds. Thefollowing theorem summarizes known results on prop-erties of valid inequalities (pagoda functions) for CB.

Theorem 3.3[3]: For each valid inequality a = [ai,j ]


Fig. 4 Three facets of the English solitaire cone.

for CB

1. For every consecutive triple T = (t1, t2, t3), theassociated triangle inequality must hold.

2. Negative coecients of a can only occur in corners.3. If T = (t1, t2, t3) is a consecutive triple with at2 =

0 then at1 = at3 .4. If two consecutive row (respectively, column) en-

tries of a are zero the entire row (respectively, col-umn) is zero.

It is not feasible to generate all facets for reason-ably sized boards, and in general no characterization offacets is known. A large class of facets can, however,be generated by the following procedure.

Genfacet /*procedure to generate a facet matrix a ofCB */

1. Choose a proper subset of coecients of a satisfy-ing: (a) If a corner is chosen, all coecients in therow and/or column of at length at least 4 contain-ing the corner must also be chosen; and (b) If twoconsecutive coecients are chosen, their entire rowand column must also be chosen. Set these chosencoecients to zero.

2. Choose any undened coecient that is not a cor-ner and set it to one.

3. Choose a consecutive triple T = (t1, t2, t3) forwhich precisely two of the corresponding coe-cients of a are dened. Dene the remaining co-ecient by the associated tight triangle inequalityproviding this does not violate any other triangleinequality for a.

4. Repeat step 3 until no further coecient of a canbe dened.

Theorem 3.4[1]: Given an m by n board B, withm 4 or n 4, if Genfacet terminates with all ele-ments of a dened, then a is a facet of CB.

Genfacet can easily be adapted to non-rectangularboards that are connected subsets of the square grid,such as the original peg solitaire game. The notion ofcorner generalizes in the obvious way to all holes thathave exactly one horizontal and vertical neighbour. For

example, the original English game has 8 corners andby Genfacet we can generate the facets (i) and (ii), butnot facet (iii), given in Fig. 4. Interesting applicationsof Genfacet include a characterization of 0-1 facets andexponential upper and lower bounds on the number offacets (in the dimension |B| of CB), see [1] for detailsand other geometric and combinatorial properties ofCB.

4. The Lattice Criterion

4.1 The Solitaire Lattice

For the fractional game, we relaxed the integrality whilekeeping the non-negativity condition. Another relax-ation of the integer game is to drop the non-negativitywhile keeping the integrality. It amounts, besides al-lowing any integer (positive or negative) number ofpegs for any intermediate congurations, to allow ad-ditive moves. The conguration of an additive move+ (jumping over an empty hole and putting a peg in)is c+ = c where c is the conguration of an ordi-nary (subtractive) move . We call this game the latticegame. Clearly, c, c is lattice feasible if and only if

c c LB =M

: Z

where the solitaire lattice LB is the set of all integerlinear combinations of moves. It gives the followingcriterion (weaker than Prop. 3.1):

Proposition 4.1 (The lattice criterion):A necessary condition for a pair of congurations (c, c)to be feasible is that c c LB.

Since the score is a homomorphism of Z-moduleswhich maps each lattice generator M to (e, e),it follows that (v) = (e, e) for any v LB; i.e., forany board B and any pair (c, c) on B if c c LBthen c c Ker(). In other words, as stated in thefollowing proposition, the lattice criterion is generallystronger than the rule-of-three.

Proposition 4.2: For any board B, we have LB Ker().


Fig. 5 An infeasible game satisfying the rule-of-three but notthe solitaire lattice criterion.

Figure 5 provides an example of a null-class board and agame on it whose associated pair (c, c) satises c c Ker() but c c LB (see Prop. 4.5). This showsthat the lattice criterion may be strictly stronger thanthe rule-of-three and therefore could be more useful inproving infeasibility.

4.2 The Lattice Criterion versus the Rule-of-Three

The solitaire lattice of a board B is typically of fullrank |B| and, in this case, let det(LB) denote his deter-minant. Since LB Ker() for any board B, we haveZB/Ker() ZB/LB and therefore, |ZB/Ker()| |ZB/LB |. For the usual board |ZB/Ker()| = 16:one can easily check that the 16 possible {0, 1}-congurations on a 2 2 board are mapped by pre-cisely onto the 16 elements of Z22. In other words, if aboard B contains a 22 sub-board then |ZB/Ker()| =|Im()| = 16. For a typical board B, the map is ontoand the lattice LB is full rank, giving |ZB/Ker()| = 16and |ZB/LB| = det(LB). This gives the following use-ful lemma.

Lemma 4.3: For a board B such that is an ontomap and LB is full rank, LB = Ker() if and only ifdet(LB) = 16.

4.3 The Lattice Criterion Beats the Rule-of-Three

A close study of LBm,n provides a canonical basis(which is precisely the Hermite basis) for this lattice,see [5] for details. We have:

Theorem 4.4: Let Bm,n be any m n board withn 4 or m 4. The solitaire lattice LBm,n hasfull rank with determinant det(LBm,n) = 16, henceLBm,n = Ker(). LBm,n is characterized by c LBm,nif and only if{

ci,0 : 0 i n 1, i 0(mod 3)} 0 (mod 2){

ci,0 : 0 i n 1, i 1(mod 3)} 0 (mod 2){

ci,1 : 0 i n 1, i 0(mod 3)} 0 (mod 2){

ci,1 : 0 i n 1, i 1(mod 3)} 0 (mod 2)

Using the basis of LBm,n , one can eciently com-pute the (Hermite normal) basis of more complexboards. The algorithm consists in covering a board B

Fig. 6 A hook board B with associated lattice satisfyingdet(LB) = 128.

by overlapping rectangular sub-boards Bi, i = 1, . . . , I.Then we append each matrix Aimi,ni i = 1, . . . , I con-sisting of the (row and column) moves within each sub-board Bi and add the cross moves, that is, the movesfrom one sub-board to another. Since the number ofcross moves is, in general, quite small, the computationof the resulting (Hermite normal) basis of the initialboard B is, in general, a bit tedious but easy. Thisalgorithm also provides a rough upper bound for thedeterminant, that is, det(LB)

Ii=1 det(LBi) 24I .

For example, the board B60 given in Fig. 5 is made oftwo overlapping square boards B6,6 and B5,5 with 4cross-moves centered on the common hole. It directlygives the following proposition which, in particular, ex-cludes any complementary game (c, c) while, B60 be-ing a null-class board, complementary games satisfy therule-of-three.

Proposition 4.5: LB60 has full rank with determi-nant det(LB60) = 32, hence LB60 = Ker(), andl LB60 , that is, any complementary game (c, c) isinfeasible on B60.

The computation the basis of LB60 can be easilyextended to any set of k rectangular boards pairwiseoverlapping on a common corner. The resulting k-hook boardsee for example Fig. 6will give a lat-tice with determinant 2k+3. Hook boards demonstratethat the solitaire lattice criterion can be exponentiallyner than the rule-of-three in the sense of the followingtheorem.

Theorem 4.6: For hook boards, the solitaire latticecondition exponentially outperforms the rule-of-three,that is, for every k and every k-hook board B, the ra-tio of the number of congruence classes of ZB moduloLB to the number of congruence classes of ZB moduloKer() satises

|ZB/LB ||ZB/Ker()| = 2

k1.

5. Conclusion

The solitaire game provides an nice application for clas-


Fig. 7 A lattice and fractional feasible but integer infeasiblegame.

sical combinatorial tools such as cone, lattice and inte-ger cone. Checking membership in the lattice LB isusually easy (once we have a basis) and checking mem-bership in the cone CB amounts to solve a linear pro-gram in polynomial time. Combining these two criteria,that is, checking membership in CB LB, is usually ef-cient in proving infeasibility. For example, while thegame in Fig. 3 satises c c LB but c c / CB,the central game (see Fig. 1) played on a French board(an English board with 4 additional holes in positions(2,2)) satises c c CB but c c / LB. Notethat for both the French and English boards, we haveLB = Ker(); therefore checking the membership in LBcan be easily done using the rule-of-three. The mem-bership in CB of the central game played on a Frenchboard can be shown by moving all pegs on the bound-ary to the inner part of the board, then moving onepeg in the center and nally remove the other pegs us-ing the fractional move given after Prop. 3.1. Clearlywe have CB LB ICB but this inclusion is strict asillustrated by Fig. 7. A further step could be to nda relatively small generating set (Hilbert basis) for theinteger cone ICB of some interesting classes of boardsB.

Acknowledgments

Research supported by N.S.E.R.C. and F.C.A.R. forthe rst author, by a Japan Society for the Promo-tion of Sciences Fellowship for the second author andin part by a grant from the Israel Science Foundation,by a VPR grant, and by the Fund for the Promotion ofResearch at the Technion for the third author.

References

[1] D. Avis and A. Deza, On the solitaire cone and its relation-ship to multi-commodity ows, Mathematical Programming(to appear).

[2] D. Avis and A. Deza, On the binary solitaire cone, DiscreteApplied Mathematics (to appear).

[3] J.D. Beasley, The Ins and Outs of Peg Solitaire, Oxford Uni-versity Press, 1992.

[4] E.R. Berlekamp, J.H. Conway, and R.K. Guy, Purging PegsProperly, Winning Ways for your mathematical plays, vol.2,Academic Press, pp.697734, 1982.

[5] A. Deza and S. Onn, Solitaire lattices, Graphs and Com-binatorics (to appear).

[6] G.W. Leibniz, Miscellenea Berolinensia ad incrementum sci-entariua, vol.1, pp.2226, 1710.

[7] R. Uehara and S. Iwata, Generalized Hi-Q is NP-complete,IEICE Trans., vol.E73, pp.270273, 1990.

[8] J.N. Vallot, Rapport sur un travail de Suremain de Mis-sery: Theorie generale du jeu de solitaire, considere comme

proble`me danalyse et de situation, Compte-rendu destravaux de lacademie des sciences, arts et belles-lettres deDijon, pp.5870, 18411842.

David Avis got his Ph.D. in Oper-ations Research at Stanford University in1977. Since then he has been a profes-sor in the School of Computer Science atMcGill University. He is a frequent visi-tor to Japan, where he has been a visitingprofessor at Chuo University, Kyoto Uni-versity, Kyushu University, Osaka ElectroCommunications University, Tokyo Insti-tute of Technology and the University ofTokyo. His research interests are in dis-

crete mathematics and geometry, especially in convex polyhedra.

Antoine Deza received his Ph.D.from the Tokyo Institute of Technologyin 1996 and his M.Sc. from Ecole Na-tional des Ponts et Chaussees, Paris. Af-ter being a research fellow in Departementde Mathematique, Ecole PolytechniqueFederale de Lausanne, Suisse, DavidsonFaculty of IE & M, Technion, Israel andCentre dAnalyse et de Mathematique So-ciales, Ecole des Hautes Etudes en Sci-ences Sociales, Paris, France, he is cur-

rently a JSPS fellow in the Department of Mathematical andComputing Sciences, Tokyo Institute of Technology. His researchinterests are in combinatorial optimization and discrete and com-putational geometry.

Shmuel Onn received his Ph.D. inOperations Research from Cornell Uni-versity in 1992. Following a researchfellowship in Combinatorial Optimizationat DIMACS, Rutgers University, and anAlexander von Humboldt research fellow-ship in Mathematics and Computer Sci-ence at the University of Passau in Ger-many, he joined in 1994 the Davidson Fac-ulty of IE&M at Technion - Israel Insti-tute of Technology. He held visiting posi-

tions at the Institut Mittag Leer in Stockholm, at the Univer-sity of Trier, and at the Mathematical Sciences Research Institute(MSRI) in Berkeley. His research concerns the structural studyof discrete optimization problems and the search for ecient al-gorithms for solving them, emphasizing the use of geometric andalgebraic methods.

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