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The Slice Group in Rubik's Cube

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The Slice Group in Rubik's Cube The Slice Group in Rubik's Cube DAVID HECKER RANAN BANERJI SainitJoseph 's Universitv Philaidelphia,PA 19131 The differentmaneuvers on Rubik'scube can be thoughtof as a setof transformationsforming a subgroupof thegroup of all permutationsof thecube's 54 facelets.In thispaper, we shalldo a completestudy of thesmall subgroup of thisgroup generated by turningonly the center layers of thecube. This is knownas theslice group. Our purpose in thisstudy is to use thecube to illustrate severalfundamental group theoretic techniques. Beforean accuratestatement of theproblem can be made,we mustintroduce some notation, mostof whichis standardin thecube literature [4]. We shalldescribe all maneuversas ifthey were carriedout withrespect to the cube held in a fixedposition, referring to the six facesby the "colors" F, B, R, L, U, and D forfront, back, right,left, up, and down. The cube will be assumedto startwith all six faceshaving solid colors,known as theclean stateof thecube (see FIGURE 1). A clockwise900 turnof any of thesesix faces(clockwise looking at theface "from outside")will be denotedby TF,TB, etc.(see FIGupE 2). We shallalso talkabout a 90? clockwise turnof theentire cube looking at thecorresponding face from outside and denotethese by CF,BC,, etc. Everypossible maneuver M can be writtenas a finitesequence of thesefundamental moves, hence thesemoves generate the group of all possibletransformations of the cube. The inverseof any move M, denotedby M1, is themaneuver required to undo theeffect of themove M. Note thatthe actual twistsof the cube neededto accomplishthis will not be unique.To avoid this problem,when speakingof a move or maneuveron the cube, we referonly to its effective permutationof the cube's 54 facelets,and not to the actual twistsrequired. Two different sequencesof twistshaving the same effectare consideredto be thesame move.Repeating any movea numberof timeswill be expressedwith the usual exponential notation. In additionto referringto thecube's facelets, we shallalso referto the27 subcubesof thecube, IB I B- I ? BB B BB IU uU U U /UU U U /U R LllI u u I L F F F ~~~R/R /U R 1R IRI RR / F F F R LF F F R R R L R F F F RR F FEF F F F DDD (a) (b) FiGuKEL 1. (a) Rubik'scube in the clean state.(b) Rubik'scube in the clean statewith hidden sides displayed. VOL. 58, NO. 4, SEPTEMBER 1985 211 This content downloaded from 65.206.22.38 on Mon, 19 May 2014 12:55:43 PM All use subject to JSTOR Terms and Conditions U F U U F F R FIGuRE 2. The moveTR. oftencalled cubiesin theliterature [4]. One of thesecubies lies in theinterior of thecube and cannot be seen. Eight othersappear on the cornersof the cube; theseeach have 3 facelets showing.Twelve are edgecubies with 2 faceletsshowing, and theremaining six lie at thecenter of each face and have onlyone faceletvisible. The cubiesare importantsince the facelets on each cubie stay on that cubie so thata large numberof the 54! permutationsin the groupof all permutationsof the 54 faceletsare prohibited.Also, everymove takescorner cubies to corner positions,edge cubies to edgepositions, and centercubies to centerpositions, thus further limiting thepossible permutations. The positionof anyfacelet or cubiewhen the cube is in theclean state is called thehome position of thatfacelet or cubie.The clean stateis completelycharacterized by all faceletsbeing in theirhome position. We thinkof thisposition as representingthe identity permutationon thecube. The slice group The slice groupis thegroup of transformationsof the cube generatedby thefollowing three movesequences: TBTF CB , TLTR1CL1, TUT-1C-1 (1) These are illustratedin FIGURE 3. One can thinkof themin twoways. First, consider them as turningtwo opposite faces "parallelly," one clockwiseand one counterclockwise,then turning the wholecube to returnthose two faces to theiroriginal positions. Alternately, one can considerthe overalleffect, which is to turnthe center slice of thecube clockwise 900 whenlooking at theF, R, and D facesrespectively. Our discussionwill be fromthis latter point of view. We shalldenote the three sequences in (1) by thesingle letter F, R, and D, whichwill stand for turns of thecorresponding center slice. We IB B IB| B B B L I R > - U /UF /F U U U u D uu u U F U R u F L R F FE F F U F U RI u U uL L D RR R RR F D F F F F DD F:TBTF 1C1 R:TLT,lC7-1 D:TuTi1Ci FIGURE3. The threemoves which generate the slice group. 212 MATHEMATICS MAGAZINE This content downloaded from 65.206.22.38 on Mon, 19 May 2014 12:55:43 PM All use subject to JSTOR Terms and Conditions have thusgiven these letters two meanings, as we have also used themto representcolors of the faces.The meaningin a giveninstance should be clearfrom the context. To discoverthe structure of theslice group G of permutationsgenerated by themoves F, R, and D, we analyze the actionsof slice moves on differentsets of cubies,thus constructing homomorphismsfrom G to knowngroups. Recall thatif H is anygroup of permutationson a set X, and if Y is a subsetof X suchthat forevery h E H and y E Y, we have h(y) E Y, thenY is said to be closedunder the action of H. When we have such a closed subset,a homomorphismis inducedfrom H into the groupof permutationsof Y: simplymap h in H to thepermutation it induceson theset Y. In our case, G is a groupof permutationson theset of cubies and facelets.We willfind subsets of cubiesthat are closedunder G, thusinducing homomorphisms of thetype just described. Returningto thecube, first notice that every corner cubie is leftfixed, or unmoved,by each slice move,thus the slice group induces the identity permutation on theset of cornercubies and theirfacelets. This is convenient,since the colors on thesecorner cubies act as benchmarksin case we forgetwhich face is front,back, etc. Next,slice movesalways move center cubies to centerpositions, hence the set of six center cubiesis closed underthe action of G. We denoteby T thehomomorphism this induces from G intoS6, thegroup of permutations on six objects(in thiscase, thecenter cubies). The imageT(G) in S6 ignoresthe actionof G on all cubiesexcept the centers. By examiningthe cube, one notes thateach slice movehas thesame effecton thecenters as some rigidmotion of theentire cube. For example,the move F takescenters to thesame positions that CF does. Therefore,T maps G into T, the groupof rigidmotions (rotations) of the cube (a subgroupof S6). The map T is actuallyonto T since T is generatedby CF, CR, and CD, and -(F)= CF, i-(R)= CR, and Ti(D) = CD. It is well knownthat T is isomorphicto S4, and thus has 24 elements.(The isomorphismwith S4 is unimportantin whatfollows. The interestedreader can constructthe isomorphismby thinkingof T as actingon theset of 4 diagonalline segmentslinking opposing cornersof a rigidcube.) The twelveedge cubiescan be partitionedinto threesubsets of foureach, EF, ER, and ED, whereeach set containsthe edge cubieson the centerslice parallelto the F, R, and D faces, respectively.By manipulatingthe cube, one noticesthat the generators F, R, and D (and hence everyslice move in G) takeedge cubiesin thesesets to edge cubiesin the same set. Hence, as above,we gethomomorphisms OF' 9kR, and OD, each mappingG intothe group of permutations of EF, ER, and ED, respectively.Again, by examiningthe cube,one noticesthat F inducesa 4-cycleon theelements of EF, whileR and D each inducethe identity permutation on EF. Hence OF (F) is a 4-cycle,OF (R) = OF (D) = id, and we see thatOF(G) is generatedby OF (F) and is isomorphicto Z4. For anyelement g of G, OF (g) is thenumber of timesmod 4 thatF appearsin any sequenceof movesgenerating g, withF-1 countedas -1 (or 3 since F-1 = F3). A similar analysiscan be made of thehomomorphisms OR and OD. We thushave fourhomomorphisms of G: one onto T and threeonto Z4. By combiningthe fourhomomorphisms, one obtainsa homomorphismfrom G into T x Z4 x Z4 x Z4 givenby g --*( T( 9),9OD ( 9),9OR ( 9),9OF ( 9)) - (2) The kernelof thishomomorphism is clearly the intersection of thekernels of thefour homomor- phismsdefining it, which is theset of elementsin G leavingall centercubies and all elementsof ED, ER, and EF fixed.Such a permutationmust leave thecube in its clean state,and is thusthe identity.Therefore, the homomorphism is injective and G is isomorphicto its image,a subgroup of TX Z4 X Z4X Z4. Next,we noticethat each generatorof theslice group acts as theidentity on twosets of edge cubies,as a 4-cycleon one set of edgecubies, and as a 4-cycleinside T, thegroup of motionsof the centers.Therefore, overall, each generatoracts as a productof twodisjoint 4-cycles, yielding an evenpermutation. It followsthat G is isomorphicto a subgroupof the even permutationsin T X Z4 X Z4 X Z4. We willshow that the image of G containsall sucheven permutations. (Note VOL. 58, NO. 4, SEPTEMBER 1985 213 This content downloaded from 65.206.22.38 on Mon, 19 May 2014 12:55:43 PM All use subject to JSTOR Terms and Conditions thatparity in T x Z4 X Z4 X Z4 is inheritedfrom S18, the groupof all permutationsof the 18 centerand edge cubies, of which T x Z4 X Z4 X Z4 is a subgroup.) Using(2), G can be identifiedas a subgroupof T x Z4 X Z4 X Z4, and T is theprojection onto thefirst coordinate of theproduct. Hence, ker T can onlycontain elements of theform (id, a, b, c) in T x Z4 X Z4 X Z4, witha + b + c evenfor the reasons discussed above. The followingmoves generatethe indicated elements: (i) RF-1DF (id,1, 1,0) (ii) RDFD-1 (id, 0, 1, 1) (iii) FDR-lD-lFD-lRD (id, 0, 0, 2). Rememberthat the last threecomponents of theseelements refer to thecyclic permutation of the edge cubiesin the D, R, and F planes,respectively.
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