CUBELIKE PUZZLES-WHAT ARE THEY AND HOW DO YOU SOLVE THEM? J.A. EIDSWICK Departmentof Mathematics and Statistics,University of Nebraska, Lincoln, NE 68588 1. Rubik'scube, Meffert's pyramid, Halpern's dodecahedron, Alexander's great dodecahedron, and thelist goes on. Suchobjects are partitionedinto smaller pieces that get mixed up whenyou turntheir faces or layers,yet theydon't fall apart.All are groupsin disguise;solving them amountsto findingalgorithms for factoring arbitrary elements into products of certaingenerators. Justwhat constitutes a "cubelikepuzzle"? Is therea universalalgorithm for solving them? These are questionssuggested by Douglas Hofstadter'sScientific American article [12]. Concerninga puzzle he calls theIncrediBall, Hofstadter writes: "I foundthat when I loosenedmy conceptualgrip on the exact qualitiesof my hard-won operatorsfor the Cube and tookthem more metaphorically, I could transfer some of myexpertise fromCube to I-ball.Not everythingtransferred, needless to say.What pleased me mostwas when I discoveredthat my 'quarkscrew'and 'antiquarkscrew'were directly exportable. Of course,it took a whileto determinewhat such an exportwould consist of. What is theessence of a move? What aspects of it are provincialand shedable?How can one learn to tell easily?These are difficultquestions for which I do nothave theanswers. I graduallylearned my way around the IncrediBall by realizingthat a powerfulclass of moves consistsof turningonly two overlapping 'circles' in a commutatorpattern (xyx'y'). I therefore studiedsuch two-circlecommutators on paper untilI foundones thatfilled all my objectives. They includedquarkscrews, double swaps and 3-cycles,which form the basis of a complete solution.In doingso I cameup withjust barelyenough notation to covermy needs, but I did not developa completenotation for the IncrediBall.This, it seemsto me, wouldbe mostuseful: a standarduniversal notation, psychologically as wellas mathematicallysatisfying, for all cubelike puzzles. It is, however,a veryambitious project, given that you would have to anticipateall conceivablevariations on thisfertile theme, which is hardlya trivialundertaking." Concerningalgorithms for solving the cube, David Singmaster[19, p. 12] writes:"... we need to proceedin two directions.First, by examiningthe cube and its group,we discoverwhich patternsare possible and, second,we show that we can achieveall possiblepatterns." The unscramblingproblem, then, is directlyrelated to theproblem of determiningthe structure of the underlyinggroup. Later [19, pp. 58-9], Singmasterdescribes this group as a subgroupof index12 of a directproduct of wreathproducts. Singmasteralso writes:"... it is a remarkablephenomenon that everyone seems to finda differentcombination of processesand strategy."A surveyof themany "how-to" books on the subjectwill confirmSingmaster's assertion. But commonthreads will be foundrunning through all of thesealgorithms. These include the commutators and 3-cyclesmentioned by Hofstadter. This articleis an attemptto put some algebraicorder into thebusiness of solvingcubelike puzzles.Ideally, an algebraictheory would unfold that would, in an elementaryway, yield highly efficientalgorithms for all suchpuzzles. The workhere is a stepin thatdirection. Our strategy, like Singmaster's,will lead to a determinationof theunderlying group structures. A descriptionof thesestructures is givenin ?6. Our startingpoint will be to put the above-mentionedcommon threads together to forma commonstrategy. The strategywill thenbe applied to variouspuzzles includingthe general J. A. Eidswick:I receivedmy Ph.D. in 1964 at PurdueUniversity under the direction of Louis de Branges.Since thenI have publishedmodestly in theareas of real analysisand topology.In 1981, I fellunder the spell of thecube and wrotethe booklet Rubik's Cube Made Easy. I also designedRubik's Cube EngagementCalendar 1982. The spell continues.My hobbies includebackpacking, gourmet cooking, and joggingenough to justifyeating. 157 This content downloaded from 65.206.22.38 on Mon, 19 May 2014 12:59:51 PM All use subject to JSTOR Terms and Conditions 158 J. A. EIDSWICK [March n x n x n cube.The efficiencyof thesealgorithms will not be a majorconsideration of thispaper, althoughsome attentionis givento this topic in ?7. It would be veryinteresting to obtain informationabout the length of theshortest possible algorithm. Group-theoretically, this amounts to calculatingthe least upper bound for the lengths of wordsrequired to expressall elementsof the underlyinggroup in termsof a certainset of generators.This is a difficultproblem about whichlittle is known.(See [11,p. 35],[19, pp. 52-3]; also [20] forrelated results.) In ?2, commonthreads of cube solutionsare summarized.In ?3, theconcept of wreathproduct is developedand examplesare giventhat illustrate the relationshipthat exists between wreath productsand cubelikepuzzles. In ?4, a generalstrategy is based on Propositions1-7. These resultsserve as keysfor solving most cubelike puzzles. Exceptions are the" twofaces puzzle" of [2, p. 768] (see Example4 below)and the"skewb" of [12,p. 20]. Applicationsin ?5 includethe cube puzzle, threedifferent partitions of the tetrahedron,two of theoctahedron, two of thedodeca- hedron,one of theicosahedron, and thegeneral n x n x n cube.The latterillustrates the essence of Proposition1. The only prerequisitefor readingthis articleis an elementaryknowledge of permutation groups.In particular,the reader does nothave to knowhow to "do thecube". For a discussionof permutations,see almostany introductoryalgebra text (e.g., [3], [10]). For a fairlycomplete treatmentof thesubject of permutationsand a glimpseat its evolution,see in order,[5], [6], [21], and [22]. For the generaltheory of groups,see, e.g., [9], [15], or [16]. A few "cube theory" referencesare includedat theend forthe interested reader. 2. Commonthreads. All intelligiblesolutions of thecube puzzleseem to have thesecommon features: (i) Two distinctsubproblems are recognized:the positioningproblem and the orientation problem.Mathematically, these relate to permutationgroups and wreathproducts, respectively. (ii) Two distinctorbits are recognized:the cornercube orbit and the edgecube orbit. (iii) A special parity-adjustingprocess is needed. (iv) Cubeletsare restoredone-by-one. (v) Processesfor restoring individual cubelets (which involve anywhere from zero to twenty quarter-turns)often involve conjugationsand commutators. A typicalsolution begins with operations that affect both orbits,but laterrestricts to those which affectonly one. Likewise,cubelets are positionedand orientedsimultaneously at the beginning,then separated later on. Thereare obviouslymany ways to do thisand therein,no doubt,lies theexplanation to Singmaster'sobserved "phenomenon". 3. Notation,wreath products. Throughout, 8 denotesa finiteset, G a permutationgroup actingon X, and F a subgroupof a wreathproduct. In ?5, 8 will be interpretedas a set of unorientedpuzzle pieces, G a group of permutationsof such pieces, and F a group of permutationsof orientedpuzzle pieces. Permutations will act on theright, other functions on the left.By an orbitof X is meanta setof theform (9= ((x) = {xg: ge G}. If (9 is aa orbit,then GI( denotesthe restrictionof G to ( and, forg E G, gIO denotesthe restrictionof g to (. We willwrite Act(g) forthe action set { x E 8(: xg x }, Sym8 and Alt8, respectively,for the symmetricand alternatinggroups on 8, S, = Sym{1,..., n}, An = Alt{l, ... , n } and Zr forthe group of integers mod r. If S is a finiteset, ISI denotesits cardinality and S` theset of functionsfrom 8 to S. If g and h are elementsof a group,then [g, h] denotes the commutatorghg-lh-1. Wreathproducts are usually studied along with semidirect products and/or group extensions as, e.g.,in [13],[15], and [18].The definitionbelow (cf. [14, p. 32]) assumesno previousknowledge of thesecompanion ideas. We mention,though, the important theorem of Kaloujnineand Krasner This content downloaded from 65.206.22.38 on Mon, 19 May 2014 12:59:51 PM All use subject to JSTOR Terms and Conditions 1986] CUBELIKE PUZZLES-WHAT ARE THEY AND HOW DO YOU SOLVE THEM? 159 (see [18, p. 100] or [13,p. 49]): Everyextension of a groupA bya groupB can be embeddedin a wreathproduct. DEFINITION. Let G and H be permutationgroups that act on .T= {1,. , n } and MY= {1, ... , r}, respectively.Then the wreathproduct of H by G, writtenH I G, is the subgroupof Sym(T x C) generatedby permutationsof thefollowing two types: 7T(g) :(i, j) (ig, j) forg E G and a(hl, . .., hn) (i0 j) (1* , ihi) forhl,..., hn ( H. We mayvisualize the situation as follows:Suppose n decksof cardsoccupy positions 1,..., n and supposethat each deck contains r cardswhich occupy levels 1,. , r. Then7r(g) permutes the decks accordingto g, maintainingcard levels,and a(hj,.. ., hn) shufflesthe decks in positions 1,..., n accordingto hl,.. ., hn,respectively, maintaining deck positions. In the applicationsto be considered,H willbe cyclic.Accordingly, we will take H = Zr for some positiveinteger r and use additivenotation for this group. Notice thatfor such groups, shufflingsamount to whatcard players call "cuts". For g,p E G and hl,..., hn,kl,..., knE H, we have = TT(g)'7(p) r(gp)? a(hj,...,hn) a(kj,...,kn) = a(h + kl,..., hn + kn), and ar(hl,..., hn)gT(g) = gT(g) a(hlg-l. .hng-1). It followsthat any elementa of H I G has a uniquerepresentation of the form T(a') a(a"), wherea' E G and a" e H. In particular,we have IH I GI = IG X Hfl = IGIIHIl'l. Henceforth,we willidentify elements a of H I G and correspondingpairs