wentySix Mo ves Sufce or Rubik Cube Daniel unkle Colle - PDF document

Twenty-SixMovesSufceforRubik'sCubeDanielKunkleCollegeofComputerScienceNortheasternUniversityBoston,MA02115/USAkunkle@ccs.neu.eduGeneCoopermanCollegeofComputerScienceNortheasternUniversityBoston,MA02115/USAgene@ccs.neu.eduABSTRACTThenumberofmovesrequiredtosolveanystateofRubik'scubehasbeenamatteroflong-standingconjectureforover25years—sinceRubik'scubeappeared.Thisnumberissometimescalled“God'snumber”.Anupperboundof29(intheface-turnmetric)wasproducedintheearly1990's,followedbyanupperboundof27in2006.Animprovedupperboundof26isproducedusing8000CPUhours.Onekeytothisresultisanew,fastmultiplicationinthemathematicalgroupofRubik'scube.Anotherkeyisefcientout-of-core(disk-based)parallelcomputationusingterabytesofdiskstorage.OnecanusetheprecomputeddatastructurestoproducesuchsolutionsforaspecicRubik'scubepositioninafractionofasecond.Workinprogresswillusethenew“brute-forcing”techniquetofurtherreducethebound.CategoriesandSubjectDescriptors:I.1.2[SymbolicandAlge-braicManipulation]:AlgebraicalgorithmsGeneralTerms:Algorithms,ExperimentationKeywords:Rubik'scube,upperbound,permutationgroups,fastmultiplication,disk-basedmethods1.INTRODUCTIONOverthedecades,shortsolutionstoRubik'scubehaveprovidedafascination—bothforresearchersintechniquesofsearchandenumerationandforhobbyists.Forresearchers,Rubik'scubeservesasawell-knownchallengeproblemagainstwhichotherwisedi-versemethodscanbecompared.In1982,SingmasterandFrey[2]endedtheirbookonCubikMathwiththeconjecturethat“God'snumber”isinthelow20's.Nooneknowshowmanymoveswouldbeneededfor“God'sAlgorithm”assuminghealwaysusedthefewestmovesrequiredtorestorethecube.Ithasbeenproventhatsomepatternsmustexistthatrequireatleastsev-enteenmovestorestorebutnooneknowswhatthosepatternsmaybe.ExperiencedgrouptheoristshaveThisworkwaspartiallysupportedbytheNationalScienceFoun-dationunderGrantsACIR-0342555andCNS-06-19616.Permissiontomakedigitalorhardcopiesofallorpartofthisworkforpersonalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforpro®torcommercialadvantageandthatcopiesbearthisnoticeandthefullcitationonthe®rstpage.Tocopyotherwise,torepublish,topostonserversortoredistributetolists,requirespriorspeci®cpermissionand/orafee.ISSAC'07,July29–August1,2007,Waterloo,Ontario,Canada.Copyright2007ACM978-1-59593-743-8/07/0007...$5.00.conjecturedthatthesmallestnumberofmoveswhichwouldbesufcienttorestoreanyscrambledpattern—thatis,thenumberofmovesrequiredfor“God'sAl-gorithm”—isprobablyinthelowtwenties.Thisconjectureremainsunproventoday.Atthetimeofthiscon-jecture,thebestknownboundswerealowerboundof17andanupperboundof52[2].Thecurrentbestlowerboundis20[7].Be-forethiswork,thebestknownupperboundwas27[5].Here,weimprovethatboundto26.Notethatinallcases,weconsideramovetobeanyquarterorhalfturnofafaceofthecube,alsoknownastheface-turnmet-ric.Wedonotconsiderthealternativequarter-turnmetric,whichdenesahalf-turntobetwomoves.Wepresentanew,algebraicapproachthatconcentratesonanal-ysisofcosetsforthecorrespondingmathematicalgroup.Thenov-eltyisbasedonacombinationof:asubgroupchainoflengthtwobasedonthesquaresubgroup(order663,552)anewfastmultiplication(requiringlessthan100nanosec-onds)ofeitherasymmetrizedcosetorasymmetrizedgroupelementbyagenerator(seeSections3,4.1and5fordeni-tions);efcientparallelcomputationusingtheaggregatebandwidthofparalleldisksfor7TBofintermediatestorage.(Theag-gregatebandwidthiscomparabletothebandwidthofasingleRAMsubsystem.)anefcientlycomputableperfecthashfunctionofthesetofsymmetrizedcosets(seeSection4.1fordenitions),andef-cientcomputationofitsinverse;andacompactrepresentationofthecosetgraphdatastructureusingfourbitsperstatetoencode1:41012states.Thefoundationofthemethodistodeterminethemaximaldis-tancefromtheidentityinboththesquaresubgroupandthecorre-spondingcosetgraph.Atheart,thecomputationissimplybreadth-rstsearchusingthe18generatorsofRubik'scube(includingsquaresandinversesofgenerators).Theout-of-corecomputationisre-quiredfortheconstructionofthecosetgraphinvolvingover65tril-lioncosets.Theproblemisreducedinspaceandtimethroughtheuseofthe48symmetriesofRubik'scube(generatedbythe24symmetriesofageometriccube,plusageometricinversion).Thesesymme-triesaregenerator-preservingautomorphismsintheiractiononthegroupofRubik'scube.Wecandenebothequivalenceclassesofgroupelements(symmetrizedgroupelements)andequivalenceclassesofcosets(symmetrizedcosets)undertheautomorphism.Thisreducesthesizeofthecosetgraphtoapproximately1.4tril-lionsymmetrizedcosets. Thepaperisorganizedasfollows.Section2brieyreviewssomerelatedwork.Section3presentstheoverallalgorithmfromahighlevel.Section4presentsbackgroundandsomebasiccon-cepts.Inparticular,thisincludesthedenitionsofsymmetrizedgroupelementandsymmetrizedcoset.Section5describesthefastmultiplicationalgorithm,alongwiththeperfecthashfunction.Sec-tion6presentsthedetailsofndingarelativelytightupperboundonthedistancetotheidentityelementofallelementsofasym-metrizedcoset.Section7presentsdetailsonthecomputationsandnalresults.2.RELATEDWORKOneapproachtondingboundsonsolutionstoRubik'scubewouldbetoproducetheentireCayleygraphforthecorrespondinggroup.Cooperman,FinkelsteinandSarawagiusedthismethodtoshowthat11movessufceforRubik's222cube[1].Forthefull333Rubik'scube,thisisnotfeasible,sinceithasover4:31019states.Therstpublishedupperboundwas52.DiscoveredbyThistleth-waite[2],itwasbasedonsolvingthecubeinaseriesoffoursteps,correspondingtoachainofsubgroupsoflengthfour.Thefourstepswereproventohaveworstcaselengthsof7,13,15,and17,forthetotalof52.In1992,thisalgorithmwasimprovedbyKociemba[3]touseasubgroupchainoflengthtwo.In1995,Reid[6]provedtheworstcaseforthetwostepswas12and18,foratotalupperboundof30.Furtheranalysisshowedthattheworstcaseneveroccurs,andsoaboundof29wasshown.ThisboundwasfurtherrenedbyRadu[5]in2006to27,whichwasthebestupperboundbeforethiswork.Besidesworkintomethodswithprovableworstcases,severaloptimalsolverswithnoworstcaseanalysishavebeendeveloped.ThemethoddevelopedbyKociembaandanalyzedbyReidhasanaturalextensionthatguaranteesoptimalsolutions.Korf[4]usedsimilartechniquestooptimallysolvetenrandomcubestates,onein16moves,threein17moves,andtheremainingsixin18moves.3.OVERVIEWOFAPPROACHThegroupofRubik'scubeisdecomposedintoachainoflengthtwo,usingthesquaresubgroup(thesubgroupgeneratedbyusingonlyhalf-turnsofthefaces)astheintermediatesubgroup.Thesquaresubgrouphasonly663;552elements,andthereareapprox-imately6:51013cosetsinRubik'sgroup.Thisisincontrasttopreviousrelatedapproaches,whichusedsubgroupsthatprovidedsubproblemsofnearlyequalsize.TheoverallstrategyforanimprovedupperboundforRubik'sCubehasthefollowingthreephases.First,inSection3.1,weproduceagraphofthesymmetrizedsubgroup(symmetrizedCay-leygraph),withsubgroupelementsasnodesandthegeneratorsasedges,andshowthatnoelementhasadistanceofmorethan13fromtheidentity.Second,InSection3.2,weproduceagraphofthesymmetrizedcosets(symmetrizedSchreiercosetgraph),andshowthatnocosethasadistancemorethan16fromthetrivialcoset.Thesersttwostepsprovideaninitialupperboundof29.Finally,inSection6,weefcientlyndstilltighterbounds,byexaminingthedistancefromgroupelementsinthesymmetrizedcosettothetrivialgroupelement,yieldingthenalupperboundof26.3.1ConstructionofsymmetrizedCayleysub-groupgraphBecauseofthesmallsizeofthesquaresubgroup,only15,752afterreductionbysymmetries,thesecomputationsarenegligiblewhencomparedtothecorrespondingcomputationsforthecosets.First,weconstructedtheCayleygraphofthesquaresubgroupbybreadth-rstsearch,usingthesquaregenerators.Thiscomputa-tionstookonlyseconds,evenusingasimpleimplementationonasinglecomputer.Then,wefoundtheoptimalsolutionforallele-mentsofthesquaresubgroup,whereweallowedanyofthe18gen-eratorstobeused.Thiswasdoneusingabidirectionalsearchforeachofthesubgroupelements,usingadayofCPUtime.Wechosetousebidirectionalsearchforthiscase,whichpro-ceedsintwophases.First,weperformedaforwardsearchtodepth7usingallofthegenerators.Then,weperformedonebackwardsearchforeachofthe15,752elementsofthesquaresubgroup.Thesebackwardssearchesrequiredanywherefrommillisecondstoafewhoursintheworstcase.Overall,thisoptimizationtooklessthanoneday,requiringnoparallelization.Table1inSection7showsthedistributionofelementsforthesquaresubgroup,bothforjustthesquaregeneratorsandwhenal-lowingallgeneratorsofthecubegroup.3.2ConstructionofsymmetrizedSchreiercosetgraphTheconstructionisessentiallyaqueue-basedimplementationofbreadth-rstsearch.Thecomplexityinthealgorithmisprimarilyduetothenecessityofreducingthesearchspacebythe48symme-triesofthecube,andofusingdisk.Theprimarydatastructureisanalmostperfectlydensehashar-rayof1:51012entries,correspondingthetherangeofhashin-dicesforallsymmetrizedcosets.Theentriesholdafourbitvaluedescribingthelevelatwhichthecorrespondingsymmetrizedcosetoccurs,foratotalarraysizeof685GB.Algorithm1describeshowthisarrayislledwiththelevelatwhicheverysymmetrizedcosetoccursinthebreadth-rstsearch.Itusesaniterativeprocesswithtwophases,generatingandmerging,withonesuchiterationperlevel.Asanoptimization,thebreadth-rstsearchisinitiallyconductedusingonlymainmemory,asinthetraditionalcase,andswitchestotheparalleldisk-basedversiononceRAMlimitationsaremet(inthiscase,atlevel9).4.NOTATIONANDBASICCONCEPTS4.1GrouptheorydenitionsWereviewtheformalmathematicaldenitions.RecallthatagroupGisasetwithmultiplicationandanidentitye(eg=ge=g),inverse(gg1=g1g=e),andanassociativelaw((gh)k=g(hk)).Apermutationofaset\nisaone-to-oneandontomappingfrom\nto\n.Compositionofmappingsprovidesthegroupmultiplication,andthegroupinverseistheinversemapping.ApermutationgroupGisasubsetofthepermutationsofaset\nwiththeaboveoperations.AsubgroupHGisasubsetHthatisclosedundergroupoperations.AgroupGhasgeneratorsSG,writtenG=hSi,ifanyelementofGcanbewrittenasaproductofelementsofSandtheirinverses.Theorderofthegroupisthenumberofelementsinit,jGj.GivenagroupGandasubgroupHG,acosetofHisthesetHg=fhg:h2Hg.AsubgroupHGpartitionsthegroupintocosets.ThesetofallcosetsiswrittenG=H.Theconjugateofgbyhisdenedbyhgdef=g1hg.NGisnormalinGif8n2N;g2G;ng2N.AnautomorphismofagroupGisaone-to-oneandontomap-pingofGsuchthatforg1;g22G,onehas(g1g2)=(g1)(g2).TheinformalideaofsymmetriesofRubik'scubehasitsformalanalogueinautomorphisms.ACayleygraphofagroupGwithgeneratorsSisadirectedgraphwhoseverticesaretheelementsofGandwhosedirectededges,(g1;g2),satisfyg1s=g2forsomeedgelabels2S.SinceourchosengeneratingsetforRubik'scubeispreservedunderin-verses,theCayleygraphofRubik'scubecanalsobeconsideredanundirectedgraph.ASchreiercosetgraphofagroupGwithgeneratorsSandsubgroupHGisagraphwhoseverticesaretheelementsofG=Handwhoseedges,(Hg1;Hg2),satisfy Algorithm1ConstructSymmetrizedSchreierCosetGraph1:Initializearrayofsymmetrizedcosetswithalllevelssettoun-known(fourbitspercoset).2:Addtrivialcosettoarray;setlevel`to0.3:whilepreviouslevelhadproducednewneighbors,atnextleveldo4:fGeneratenewelementsfromthecurrentlevelg5:Letasegmentbethosenodesatlevel`amongNconsecutiveelementsofthearray.6:Scanarraystartingatbeginning.7:whilewearenotattheendofthearray,extractnextsegmentofarrayanddo8:foreachnodeatlevel`(representingasymmetrizedcoset)do9:foreachgeneratordo10:Computeproductbyfastmultiplication.11:Computehashindexofproduct.12:Savehashindexinbucketb,wherebisthehighbitsofthehashindex.Note,weonlysavethelowor-derbitsofthehashindexnotencodedbythebucketnumber.(Thisvaluetsinfourbytes.)13:Ifbucketbisfull,transferit(writeit)toadiskleforbucketb.14:endfor15:endfor16:Transferallbucketstocorrespondingdiskles.17:endwhile18:fNowmergebucketsintoarrayofsymmetrizedcosets.g19:foreachbucketbondiskdo20:Loadportionoflevelarraycorrespondingtobucketbintomainmemory.21:foreachbufferedelementondiskforthisbucket(readinlargechunks)do22:Lookupcorrespondinglevelvalueinarray.23:Ifavaluealreadyexistsfortheelement,itisadupli-cate.Otherwise,setitslevelto`.24:endfor25:Writeportionoflevelarraybacktodisk.26:endfor27:Incrementlevel`.28:endwhileHg1s=Hg2forsomeedgelabels2S.Symmetrizedgroupelementsandsymmetrizedcosets.Whiletheabovedenitionsarestandard,weextendthemtosym-metrizedgroupelementsandsymmetrizedcosets.LetAbeagroupofautomorphismsofagroupG.AsymmetrizedgroupelementgAforg2Gisthesetf(g):2Ag.GivenasubgroupHG,letAbeagroupofautomorphismsofGthatalsopreserveH(8h2H;2A;(h)2H).Thesym-metrizedcosetHgAisthesetofelementsfh(g):h2H;2Ag=[2AH(g).(Note(HgA)=HgA82A.)LetAbeagroupofautomorphismsofG.AssumethatApre-servesthegeneratingsetSofG.ThenthesymmetrizedCayleygraphfor(G;S;A)isthedirectedgraphwithverticesfgA:g2Ggandwithedges(gA1;gA2)satisfyingforsomeedgelabels2S:8g012gA1,g01s2gA2.Withoutlossofgenerality,itsufcestoconsideronlyedgelabelssatisfyingg1s2gA2foradistinguishedgroupelementg12gA1.NotethattheedgelabelinasymmetrizedCayleygraphisnotuniquesincewecouldequallywellconsider(g01)(s)2(g2)asdeningthatedge.Hence,theedge(gA1;gA2)hasedgelabelssuchthatg1s=g2andalsoedgelabels0=(s)forsome2A,suchthat(g1)(s)=(g2).Foranyelementg01s2gA2satisfy-ingg012gA1,thereisan2Awith(g01)=g1andsotheedge((g01)A;(g01s)A)=(gA1;(g1(s))A).Similarly,assumingthatG=hSi,HG,andApreservesS,onedenesthesymmetrizedSchreiercosetgraphfor(G;H;S;A)asthedirectedgraphwithverticesfHgA:g2Ggandwithedges(gA1;gA2)satisfyingHg012HgA1andHg1s2HgA2.Asbefore,inordertondallneighborsofgA1inasymmetrizedCayleygraph,itsufcestochooseanydistinguishedelementg02HgA1,andthesetofneighborsisf(H(g0)A;H(g0s)A):s2S;g0s=2H(g0)Ag:Notethatforidentitye2G,thetrivialcosetHe=HandthetrivialsymmetrizedcosetHeA=Hareequal.Forourwork,werequirethefollowingproperty:Letg02HgA.Thenthedistancefromthesym-metrizedcosetHgAtothetrivialsymmetrizedcosetHinthesymmetrizedSchreiercosetgraphisthesameasthedistancefromthecosetHg0tothetrivialcosetHintheSchreiercosetgraph.Thepropertyiseasytoprove.Ifthereisawordw0withdistin-guishedelementg002HgAsuchthatg00w02HeA=H,thenthereisan2Awith(g00)=g0,andthereforeg0(w0)=(g00)(w0)2H.SincetheautomorphismspreserveS,(w0)isawordinSoflengthd,fromHg0tothetrivialcosetintheSchreiercosetgraph.Perfecthashfunction.Next,aperfecthashfunctionisahashfunctionthatproducesnocollisions.Henceitisone-to-one.Section5describesefcientper-fecthashfunctionsbothforcertainclassesofsymmetrizedcosetsandforsymmetrizedgroupelements.Thereisanefcientlycom-putableperfecthashfunctionforsymmetrizedcosetsofourchosensubgroupofRubik'sgroup.Whilethisperfecthashfunctionisnotminimal,itisnearlyso.Furthermore,ithasanefcientlycom-putableinversehashfunction.4.2Rubik'scubedenitionsWeassumethereaderhasseenaRubik'scube,andweprovidethisdescriptionsolelytoxterminologyaccordingtostandardcon-ventions[2].ARubik'scubeisbuiltfrom26cubies,eachabletomakere-strictedrotationsaboutacoreofRubik'scube.AfaceofRubik'scubeisaside.Eachfaceisdividedinto9facelets,whereeachofthe9faceletsispartofadistinctcubie.Acubieiseitheranedgecubie(twovisiblefacelets),acornercubie(threevisiblefacelets),oracentercubie(onevisibleface,inthecenterofaside).Thefaceletsaresimilarlyedgefacelets,cornerfacelets,orcenterfacelets.ThestatesofRubik'scubecanbeconsideredaspermutationson48facelets(the24cornerfaceletsandthe24edgefacelets).Thecenterfaceletsareconsideredtobexed,andallrotationsofRu-bik'scubeareconsideredtopreserveaxedorientationofthecubein3dimensions.ThehomepositionorsolvedpositionofRubik'scubeisoneinwhichallfaceletsofafacearethesamecolor,and(forthesakeofspecicity)thebluefaceisdownwards.WewillspeakinterchangeablyaboutanelementofRubik'sgroup,astate,orapositionofRubik'scube.Similarly,wewillspeakinterchange-ablyaboutthehomepositionofRubik'scubeortheidentityelementofRubik'sgroup.Hence,apositionofthecubeisidentiedwithagroupelementthatpermutesthefaceletsfromthehomepositiontothegivenposition.ThemovesofRubik'sgroupareconventionallydenotedU,U1,U2,D,D1,D2,R,R1,R2,L,L1,L2,F,F1,F2,B,B1,andB2.Mnemonically,Ustandsforaclockwisequarterturnof the“up”face,andsimilarly,D,R,L,FandBstandfor“down”,“right”,“left”,“front”and“back”,respectively.Anyoftheabovemoveswillmoveexactly20facelets.ThesemovesalsomakeupthegeneratorsoftheRubik'scubegroup,G.Standardtechniques,suchasSims'salgorithmforgroupmembership,showthattheorderofRubik'sgroupisjGj=43;252;003;274;489;856;000(approxi-mately4:31019).ThegeneratorsofthesquaresubgrouparegivenbyQ=\nU2;D2;R2;L2;F2;B2StandardtechniquesshowthattheorderofthesquaresubgroupisjSj=663;552(approximately6:6105).Theindex,ornumberofcosets,is[G:S]=jGj=jSj=65;182;537;728;000(approx-imately6:51013).4.3Symmetries(NaturalAutomorphisms)ofRubik'sCubeForRubik'scube,wedesireasubgroupof48automorphismsthatpreservethesetofgenerators:thenaturalautomorphismsofRubik'scube.Eachautomorphismcanbeidentiedwithasymme-tryofageometriccube:eitheroneof24rotationsoftheentirecubeorarotationfollowedbyaninversionofthecube.Aninversionofthecubemapseachcornerofthecubetotheoppositecorner.ArotationofthecubemapsthenaturalgeneratorsofRubik'scubetogenerators.Aninversionofthecubemapsgeneratorstoinversegenerators(clockwisequarterturnstocounter-clockwisequarterturns).These48symmetriesofthecubeareknowntopre-servethenaturalgeneratorsofRubik'sgroup,andnootherauto-morphismsofRubik'sgroupdoso.5.FASTGROUPMULTIPLICATIONINTHEPRESENCEOFSYMMETRIESNext,themethodoffastmultiplicationispresented.Themethodbreaksupagroupintosmallersubgroups(calledcoordinatesbyKociemba[3]).Thechosensubgroupsaretailoredforfastmul-tiplicationbothwithinthesquaresubgroup(groupgeneratedbysquaresofgenerators)andmultiplicationofthecosetsbygenera-tors.Eachsubgroupissmallenoughsothatthederivedtable-basedcomputationstinsideCPUcache.Wewillseparatelyconsidertheedgegroup(theactiononedgefacelets)andthecornergroup(theactiononcornerfacelets).Thetwoactionsare“linked”.Thiswillbediscussedinlatersubsec-tions.First,weprovidethefundamentalbasisforourresult.5.1DecompositionintoSmallerSubgroupsandFastMultiplicationThefollowingeasyresultprovidesthebasisforourfastmulti-plication.LEMMA1.LetGbeagroupwithG�H=QNandQ\N=f1g.ThengivenasetofcanonicalcosetrepresentativesofG=H,anelementg2Guniquelydenesq,gandnasfollows.g=qgn;whereq2Q,gisthecanonicalcosetrepresentativeofHg,andn2N.PROOF.Clearly,g=hgforauniquelydenedh2HandgthecosetrepresentativeofHg.Theequationh=qn0thenuniquelydenesqandn0forq2Qandn2N.Denen0byn=n0g.Theng=hg=qn0g=qgn.Lemma1providesthebasisforaperfecthashfunctionforGandQ,sincegandthecorrespondingcosetQgcanbeencodedby1and2asfollows.Forg=qgnasabove;1(g)=(q;g;n)(1)and2(Qg)=(g;n):(2)GivenperfecthashfunctionsforQ,G=HandN,theaboveequa-tionsyieldperfecthashfunctionsforGandforG=Qthroughamixedradixencoding.Theimportanceofthisparticularencodingisthatitadaptswelltoanalgorithmforfastmultiplication.IfthegroupNisalsonormal,thenthereisanefcientmul-tiplicationofatriple(q;g;n)andapair(g;n).Recallthattheconjugateofagroupelementhbygisdenedashg=g1hg.RecallthatasubgroupNGisanormalsubgroupif8n2N,andg2G,ng2N.Thisimmediatelyimpliesng=gngandngng2=(n1n2)gforn;n1;n22Nandg2G.Next,considerg=qgnasabove.Inaddition,assumethatNisanormalsubgroup.Letgsdecomposeintoq0gsn0byLemma1,wheregsisthecanonicalcosetrepresentativeofHgs.NotealsothatHgs=Hgsimpliesgs=gsforgsthecanonicalcosetrepresentativeofHgs.Thentheproductofgbyageneratorscanbeexpressedasfollows.LEMMA2.LetQGandletNbeanormalsubgroupofG.LetQ\N=f1g.Letg;s2G.Assumethedecompositionsg=qgnandgs=q0gsn0,asgivenbyLemma1.Thenthefollowingholds.Ifg=qgnandgs=q0gsn0;thengs=(qq0)gs(n0ns)(3)andQgs=Qgs(n0ns):(4)ProvidingthatQ,NandG=(QN)aresufcientlysmall,Lemma2providesthefoundationforprecomputingsmalltablesofallthere-quiredproducts,foranygivens2G.Typically,wechoosestobeageneratorsincetherearefewgenerators,andsotherelativelysmallsizeofthetablesismaintained.Fastmultiplicationbyanarbitrarygroupelement,whileusingsmalltables,isalsofeasible.ByLemma1,anarbitrarygroupele-mentcanbewrittenasg=qgn.Assumingthatgisrepresentedasthetriple(q;g;n),onecansuccessivelytreateachofq,gandnasthegeneratorsforpurposesofmultiplicationontheright.5.2FastMultiplicationofCosetbyGeneratorforEdgeGroupFirst,considertheedgegroupE,therestrictionofRubik'sgrouptoactonlyonedgefacelets.Similarly,letMEbetherestrictionofthegeneratorsofRubik'scube,M,toelementsthatactonlyontheedges.LetQEbetherestrictionofthesquaresubgroupQ=\nfs2js2Mgtoactonlyontheedges.Hence,QE=\nfs2js2MEg.LetNEbethenormalsubgroupofEthatxestheedgecubiessetwise,butallowsthetwofaceletsofanedgecubietobetrans-posed.Thereare12edgecubies,butitisnotpossibleinRubik'scubetotransposethefaceletsofanoddnumberofedgecubies.GrouptheoreticargumentsthenshowtheorderofthegroupNEtobe211.ItiseasytoshowthatNEisnormalinGsinceforn2NEandg2G,g1ngmaypermutethecubiesaccordingtog1,butnxesthecubies,andgthenbringsthecubiesbacktotheiroriginalposition.So,g1ng2NandsoNEisnormal.WedescribethemethodforfastmultiplicationofacosetofE=QEbyageneratorfromME.WedeneHE=QENE.ThemethoddependsonasubgroupchainE�HE�NE;forNEnormalinE;HEdef=QENE;andQE\NE=f1g:Fortheremainderofthissection,weoftenomitthesubscriptsofME,HE,NEandQE,sincewewillalwaysbeconcernedwiththeactiononedges.ByLemma1,foranycosetQg2E=Q,Qg=Qr1r2.So,Qgisrepresentedasapair(r1;r2)forr1acanonicalcosetrepresentativeinE=Handr22N.GivenageneratorsofE,equation4is EdgeTablesSizeInputsOutputTable1a1564182Br1;sHr1s2E=Hforr1sacanonicalcosetrepresentativeofE=HTable1b1564182Br1;sr2def=nr1s2N,wherenisde®nedbysettinghdef=r1s(r1s)12Handuniquelyfactoringh=qnforq2Q,n2NTable22048182Br2;srs22NLogicalop'sr2;r02r2r022N(usingadditionmod2onpacked®elds)Figure1:EdgetableforfastmultiplicationEdgeTablesSizeInputsOutputTableAut1564181Br1;e;s2Afor(r1s)acanonicalcosetrep.ofHinE(Wechoosesuchthat(r1s)=min2A(r1s).)Table1a1564182Br1;e;sH(r1s)2E=Hforde®nedintermsofr1andsbyTableAut.(cosetrep.)(NotethatHA=H.)Table1b1564182Br1;e;sr2def=n0(r1s)2N,whereh0def=(r1s)(r1s)12Handh0=q0n0(N)forq02Q,n02Nforde®nedintermsofr1andsbyTableAutTable22048182Br2;e;srs22NTable52048482Bne2N;(n)2N,whereistheoutputofTableAutnisde®nedbyn=rs2(outputofTable2foredges)Logicalop'sr2;e;r02;er2r022N(usingadditionmod2onpacked®elds)Figure2:Edgetableforfastmultiplicationofsymmetrizedcosetbygeneratorusedbelowtomultiplythepair(r1;r2)bysandreturnanewpair,(r01;r02)=(r1s;r2(rs2)).Letr1s=qr1sr2;whereq2Q;r22N;andr1sisthecanonicalcosetrepresentativeofQr1s.(5)ThenQr1r2s=Qr1s(rs2)=Qr1s(r2(rs2))(6)Given(r1;r2)andagenerators,onecancompute(r01;r02)suchthatQr1r2s=Qr01r02primarilythroughtablelookup.Figure1describesthenecessaryedgetables.Notethatthelogicaloperationscanbedoneefciently,becauseNisanelementaryabelian2-group.ThismeansthatthegroupNisisomorphictoanadditivegroupofvectorsoveraniteeldoforder2.Inotherwords,multiplicationinNisequivalenttoaddi-tioninGF(2)11,the11-dimensionalvectorspaceovertheeldoforder2.Additionintheeldoforder2canbeexecutedby“ex-clusiveor”.Hence,itsufcestousebitwise“exclusiveor”over11bitsforgroupmultiplicationinNE.5.3ExtensiontoGroupActiononCornersForcorners,weusethesamelogicaspreviously,butwiththecornergroupC,therestrictionMCofthegeneratorstocorners,andtherestrictionQCofthesquaregrouptocorners.LetNCbethesubgroupofCthatxesinpositionthecornercubies,butallowsthefaceletsofthecornercubietobepermuted.Thereare8cornercubies,butastandardgroup-theoreticalgorithmshowsthatwithinthesubgroupC,thenumberofgroupelementsxingallcornercubiesisonly37.Asbefore,wedropthesubscriptsforreadability.Hence,thetablesofFigure1canbereinterpretedaspertainingtocorners.However,thereisasmalldifference.SinceNCisanelemen-taryabelian3-group,multiplicationinNCisequivalenttoadditionoverGF(3)7.5.4GeneralizationtoFastMultiplicationoverSymmetrizedCosetsAssumethattheautomorphismgroupAactsonedgefaceletsandcornerfacelets,separatelypreservingedgeandcornerfacelets.AssumealsothatApreservesthesubgroupsQ,NandH=QN.Therefore,AalsomapstheprojectionsofQ,NandH=QNintoedgesandintocorners.AssumethatthesymmetrizedcosetQgAisuniquelyrepresentedasQ(r1r2)A,wherer1isthecanonicalrepresentativeofasym-metrizedcosetHgAwithH=QN,andr22N,asdescribedinLemma1.Thesubscripte,below,indicatestherestrictionofapermutationtoitsactiononlyonedges.Similarly,thesubscriptcisforcorners.Thesubscriptsareomittedwherethemeaningisclear.Foredges,Q(r1;er2;es)=Q(r1s)(rs2)=Q(r1s)(r2(rs2));where(r1s)denedbyTable1aforedges;whereischosentominimize(r1s);r2denedbyTable1bforedges,etc.(7)Howeverforcorners,Q(r1;cr2;cs)=Q(r1s(r2(rs2)))=Q(r1s)(r2(rs2))=Q(r1s)n(r1s)(r2(rs2));whereischosenasinequation7;wherer1sdenedbyTable1a;(r1s)denedbyTable4a,n(r1s)denedbyTable4bforcorners,r2denedbyTable1b,rS2byTable2forcorners,etc.(8)Notethatthechoiceof2Aforedgesabovedependsontherebeingauniquesuchautomorphismthatminimizes(r1s).Infact,thisisnottrueinabout5.2%ofcasesforrandomlychosenr1ands.Theseunusualcasescanbeeasilydetectedatrun-time,andaddi-tionaltie-breakinglogicisgenerated.Weproceedtodescribeta-blesforfastmultiplicationforthecommoncaseofunique2Aminimizing(r1s),anddiscussthetie-breakinglogiclater.Thetablesthatimplementtheaboveformulasfollow.Whileitismathematicallytruethatwecansimplify(r1s)into(r1s),weoftenmaintainthelongerformulatomakecleartheoriginsofthatexpression,whichisneededforanimplementation.Asbefore,thesubscriptseandcindicatetherestrictionofapermutationtoitsactiononlyonedgesandonlyoncorners.Figures2and3describethefollowingedgetables,amongothers.Ideally,onewoulduseonlythesimplerformulaandtablesforedges,andcopythatlogicforcorners.Unfortunately,thisisnot CornerTablesSizeInputsOutputTable1a420182Br1;c;sHr1s2C=Hforr1sacanonicalrep.ofacosetofC=HTable1b420182Br1;c;sr2def=n0r1s2N,wheren0isde®nedbysettinghdef=r1sr1s12Handuniquelyfactoringh=qn0forq2Q,n02NTable22187182Br2;c;srs22NTable4a420482BHr1;cs2C=H;H(r1s)2C=H,whereHr1sistheoutputofTable1a,(cosetrep.)andistheoutputofTableAutonedgesTable4b420482BHr1;cs2C=H;n(r1s)2N,whereHr1sistheoutputofTable1a,(N)andnisde®nedbysettingh=(r1s)(r1s)12H,anduniquelyfactoringhintoqnforq2Q,n2NTable52187482Bnc2N;(n)2N,whereistheoutputofTableAutonedgesnde®nedbycomputingr2=n0r1s(asinTable1b),andrs2computedasinTable2,andr2rs2computedbylogicalop'soncornersLogicalop'sr2;c;r02;cr2r022N(usingadditionmod3onpacked®elds)Figure3:Cornertableforfastmultiplicationofsymmetrizedcosetbygeneratorpossible.Wemustchoosearepresentativeautomorphism2Aforpurposesofcomputation.Wechoosebasedontheprojec-tionr1;eofr1intoE(actionofr1onedges).Hence,Tables1aand1bforedgestakeinputr1ands,thencomputeasaninter-mediatecomputation,thenreturnH(r1s).Asimilarcomputationforcornersisnotpossible,becausetheintermediatevaluede-pendsonr1;eandnotonthecorrespondingelementofthecornergroupr1;c.Tie-breakers:whentheminimizingautomorphismisnotunique.TableAutintheprevioustableforedgesdenesanautomor-phismthatminimizes(r1s).Unfortunately,thereisnotalwaysauniquesuch.Insuchcases,oneneedsatie-breaker,sincedif-ferentchoicesofwillingeneralproducedifferentencodings(dif-ferenthashindices).Foreachpossiblevalueof(r1s),withchosentominimizetheexpression,weprecomputethestabilizersubgroupBAdenedbyB=f2A:((r1s))=(r1s)gweusetheformulasandadditionaltablebelowtondtheunique2Bsuchthattheproductminimizestheedgepairre-sult(r01;e;r02;e).Whereeventhisisnotenoughtobreakties,wecomputethefullencoding,whiletryingallpossibletyingautomor-phisms.Thislattersituationarisesinonly0.23%ofthetime,anddoesnotcontributesignicantlytothetime.ThetablesofFigure4sufceforthesecomputations.Foredges,Q((r1;er2;es))=Q((r1s))((rs2))=Q((r1s))((r2(rs2)))=Q(r1s)r02((r2(rs2)));where(r1s)denedbyTable1aforedges;whereischosentominimize(r1s);and2AsatisesQ((r1s))=Q(r1s)and((r1s))=(r1s)r02(r02denedinTable3);andr2denedbyTable1bforedges.(9)Howeverforcorners,Q((r1;cr2;cs))=Q((r1s(r2(rs2))))=Q((r1s))((r2(rs2)))=Q((r1s))n((r1s))((r2(rs2)))whereandarechosenasinequation9;andotherquantitiesbasedonthepreviousCornerTablesusing.(10)Table1cisimplementedmoreefcientlybystoringtheelementsofeachofthepossible98subgroupsoftheautomorphismgroup,andhavingTable1cpointtotheappropriatesubgroupBA,stabilizingr1;e;s.5.5OptimizationsInthediscussionsofar,weproducetheencodingorhashindexofagroupelementbasedonanencodingoftheactionofthegroupelementonedges,alongwithanencodingoftheactionofthegroupelementoncorners.Wecancutthisencodinginhalfduetoparityconsiderations.ConsidertheactionofRubik'scubeonthe12edgecubiesandthe8cornercubies,ratherthanonthefacelets.Wedenetheedgeparityofagroupelementtobetheparity(evenorodd)initsactiononedgecubies.(Recallthattheparityofapermutationisoddorevenaccordingtowhetherthepermutationisexpressibleasanoddorevennumberoftranspositions.)Thecornerparityissimilarlydened.Theedgeandcornerparityofasymmetrizedcoset,HgA,arewell-dened,andarethesameastheedgeandcornerparityofg.ThisissobecauseH=QN,andelementsofQandNhaveevenedgeparityandevencornerparity.Parityisunchangedbytheactionofanautomorphism.ForRubik'scube,thenaturalgeneratorshavetheedgeparityequaltothecornerparity.Sothispropertyextendstoallgroupel-ements,andhencetoallsymmetrizedcosetsHgA.Therefore,ourencodingcanassumethatedgeandcornerparitiesofsymmetrizedcosetsareequal.Thesizeofthecorrespondinghashtableisthusreducedbyhalf.NearlyMinimalPerfectHashFunction.Ifweweretoonlyusecosetsinsteadofsymmetrizedcosets(noautomorphism),thentheperfecthashfunctionthatwehavede-scribedimplicitlywouldalsobeaminimalperfecthashfunction.However,thereareexamplesforwhichQ(g)=Qgfornottheidentityautomorphism.AcomputationdemonstratesthattheperfecthashfunctionofSection5.4withtheadditionoftheparityoptimizationhasanef-ciencyof92.3%.Infact,wecomputethatthereare1;471;074;877;4401:51012jGj=jQj=44:3symmetrizedcosets.Theratio44.3/48yieldsourefciencyratioof92.3%.Furtherdetailsareomittedduetolackofspace.5.6FastMultiplicationinSquareSubgroupThereisasimilaralgorithmforfastmultiplicationinthesquaresubgroup,whichisomittedduetolackofspace. EdgeTablesSizeInputsOutputTableMultAut48481B;theproduct2ATable1c(A)1564181Br1;e;sf2A:((r1s))=(r1s)gTable3(N)2048482BHr1;cs2C=H,r02def=n00(r1)2N,wheretakenfromTable1c,andwhereh00def=(r1)(r1)12H,andh00=q00n00,forq002Q,n002NFigure4:Edgetableforfastmultiplicationofsymmetrizedcosetbygenerator,adjustedtobreakties6.“BRUTEFORCE”UPPERBOUNDSONSOLUTIONSWITHINACOSET6.1GoalofBruteForcingCosetsHavingconstructedtheSchreiercosetgraph,onewishestotestindividualcosets,andprovethatallgroupelementsofthatcosetareexpressibleaswordsinthegeneratorsoflengthatmostu.Hence,uisthedesiredupperboundwewishtoprove.RecallthatGisthegroupofRubik'scube,andQisthesquaresubgroup.ConsideracosetQgatalevel`(distance`fromthehomeposition,oridentitycoset,inthecosetgraph).Letdbethedi-ameterofthesubgroupQ.Foranygroupelementh2Qg,clearlyitsdistancefromtheidentityelementisatleast`andatmost`+d.Wedescribeacomputationtoproduceanerupperboundonthedistanceofanyh2Qgfromtheidentityelement.BecauseoursubgroupQ(thesquaresubgroup)isofsuchsmallorder,itisevenfeasibletosimplyapplyanoptimalsolvertoeachelementofacoset.Iftheoptimallengthwordsforeachelementisalwayslessthanorequaltou,thenwearedone.However,wewillpresentamoreefcienttechnique,whichcanscaletomillionsofcosets.Forsimplicity,werstassumethatwearenotapplyinganysym-metryreductionsusingtheautomorphismgroup.So,eachcosetcontains663,552elements,justasdoesthesquaresubgroup.6.2BasicAlgorithmNotethatforthecosetQg,therecanbemanypathsinthecosetgraphfromtheidentitycosettoQg.Intermsofgrouptheory,therearemultiplewords,w1,w2;:::,whereeachwordisaproductofgeneratorsofRubik'sgroup,andQw1=Qw2=:::=Qg.Notethatingeneral,thewordsaredistinctgroupelements:w16=w2,etc.Nevertheless,w1w122Q.Thisisthekeytondingarenedupperbound.Next,supposeourgoalistodemonstrateanupperboundu,for`u`+d.Letdist(q)denotethedistancefromanelementq2QtotheidentityelementintheCayleygraphofGwiththeoriginalRubikgenerators.LetQubefq2Qjdist(q)ug,thesubsetofQatdistancefromtheidentityatmostu.Next,considerQkgdef=fqgjq2Qkg.AssumethatthewordswiareoflengthdinthegeneratorsofRubik'sgroup,andthatQw1=Qw2=:::=Qg.NotethatforallelementsofQkw1,thereisanupperbound,k+d.Similarly,forallelementsofQkw2,thereisanupperbound,k+d.Therefore,theelementsofQkw1[Qkw1haveanupperboundofk+d.Morecompactly,dist(Qkw1[Qkw2)k+dMoregenerally,forwiawordinthegeneratorsofG,letlen(wi)bethelengthofthatword.Thendist(Qk+dlen(w1)w1[Qk+dlen(w2)w2)k+dsincethelengthofanywordinQk+dlen(w1)w1isatmost(k+dlen(w1))+len(w1)=k+dandsimilarlyforw2.DenethecomplementQ0def=QnQj.Wecannowwrite:Q0+dlen(w1)w1\Q0k+dlen(w2)w2fh2Qgjdist(hg)�k+dgClearly,theaboveequationcanbegeneralizedtotheintersectionofmultiplewords,w1,w2;:::.\iQ0+dlen(wi)wi=;=)8i;dist(Qwi)=dist(Qg)k+dFinally,forpurposesofcomputation,Algorithm2,below,cap-turestheseinsights.Algorithm2CosetUpperBoundInput:asubgroupQofagroupG,acosetQg;adesiredupperbound`;andasetofwordsh1;h2;:::ingeneratorsofGsuchthatQghi=Q.Output:ademonstrationthatallelementsofQghavesolutionsoflengthatmost`orelseasubsetSQgsuchthatallelementsofQgnSareknowntohavesolutionsoflengthatmost`.1:Letk=`len(g).LetU0=fq2Qjlen(q)�kgQ.Then(QnU0)gisthesubsetofelementsinthecosetQgwhichareknowntohavesolutionsoflengthatmost`.ThesetU0gisthe“unknownset”,forwhichwemustdecideiftheyhavesolutionsoflength`orless.2:Foreachi1,letUi=Ui1nfq2Ui1jdist(qghi)`len(hi)g.(Notethatqghi2Q).Bydist(qghi),wemeantheshortestpathinthefullsetofgeneratorsofG.Ifdist(qghi)`len(hi),thenqghasasolutionoflengthatmost`.Thesolutionforqgisgivenbyapathlengthlen(hi)followedbyapathoflength`len(hi)=dist(qghi).3:IfUi=;forsomeij,thenwehaveshownthatallelementsofQghavesolutionsoflengthatmost`.IfUj6=;,thenwehaveshownthatallelementsof(QnUj)ghavesolutionlengthatmost`.Forpurposesofimplementation,notethatghi2Q.So,forq2Q,qghicanbecomputedbyfastmultiplicationwithinthesubgroupQ.6.3UsingSymmetriesWenowgeneralizethemethodoftheprevioussectiontotakeaccountofreductionsthroughsymmetries.First,notethatforasymmetrizedcosetwithrepresentativecosetQgandforanaturalautomorphism,dist(Qg)=dist(Q(g)).(ThiswasdemonstratedinSection3.2).Furthermore,foranyh2Qg,dist(h)=dist((h)).Fromthis,itisclearthatanyupperboundonthedistanceofelementsinQgfromtheidentitywillalsoholdforQ(g).So,itsufcestodetermineupperboundsforasinglerepresentativeofeachautomorphismclass.Finally,onemustdeterminewhetherhw1i=2Q0+dlen(wi).Thiscanbedonebymaintainingahashtablemappingallelementsq2Qtodist(q).7.EXPERIMENTALRESULTSOurexperimentalresultshaveproventhat26movessufceforanystateofRubik'scube.Thiswasachievedinthreesteps:provingthatallelementsofthesquaresubgrouparewithin13oftheiden-tity;provingthatallcosetsarewithin16ofthetrivialcoset;and, reningtheboundonthefarthestcosetsbybruteforce,reducingtheboundby3.7.1SquareSubgroupElementsarewithin13oftheIdentityRecall,fromSection3.1,thefollowingtwo-stepprocessforthiscomputation.First,weconstructedtheCayleygraphofthesub-groupbybreadth-rstsearch,usingthesquaregenerators.Then,thedistanceforeachoftheseelementsfromtheidentity,whenal-lowingallgeneratorsofthefullgroup,weredeterminedusingbidi-rectionalsearch.Allofthesecomputationsweredoneonasinglecomputerinunderaday.Table1showsthedistributionofelementdistancesinthesquaresubgroup,usingeitherthesquaregeneratorsorthefullsetofgen-erators.SquareGeneratorsAllGeneratorsDist.EltsDist.EltsDist.EltsDist.Elts01812580181871119262711940932210409422105394351141373511277441812223141812620556135485621346162141146214748215167693Total15752Total15752Table1:Distributionofelementsinthesquaresubgroup,afterreductionbysymmetries.7.2Cosetsarewithin16oftheTrivialCosetThedominanttimeforourcomputationswasinproducingthesymmetrizedSchreiercosetgraphforthesquaresubgroupinthegroupofRubik'scube,asdescribedinSection3.2.ThecomputationusedtheDataStarclusterattheSanDiegoSu-percomputerCenter(SDSC).Weused16computenodesinpar-allel,eachwith8CPUsand16GBofmainmemory.Forout-of-corestorage,weusedDataStar'sattachedGPFS(IBMGeneralParallelFileSystem).Weusedupto7terabytesofstorageatanygiventime,asabufferfornewlygeneratedstatesinthebreadth-rstsearch.Thenaldatastructure,associatinga4-bitvaluewitheachsymmetrizedcoset,usedapproximately685GB.Thecomputationrequired63hours,orover8000CPUhours.Thefastmultiplicationalgorithmallowedustomultiplyasym-metrizedcosetbyageneratorataratebetween5and10milliontimespersecond,dependingonthesizeofavailableCPUcaches.Table2showsthedistributionofdistancesforcosetsinthesym-metrizedSchreiercosetgraph.Dist.ElementsDistanceElements019807411178:1107111010288693181:01092311127871763551:31010323121403523572991:410114241137814153183417:8101153002144219802136794:21011638336153300368643:31087490879161786298864Total13579815443401:361012Table2:Distributionofsymmetrizedcosetsofthesquaresub-group.7.3BruteForcingof3LevelsKociemba'sCubeExplorersoftware[3]wasusedtoshowthatallcosetsatlevel3havesolutionsoflengthatmost14.Todothis,itsufcedtoanalyzetheelementsatlevels12and13fromthesquaresubgroup.DenotingthesetofthosesubgroupelementsS,anddenotingalloftheelementsatlevel3inthecosetsbyT,weconsideredallpairwiseproductsST.Thereare(620+4)23=14;352suchelements.CubeExplorerwasrunoneachsuchelement.Thisprovesthatforacosetatcosetlevelx�3,(x2)+13movessufce.Combiningthiswiththeexpecteddepthofx=16forthesymmetrizedSchreiercosetgraphyieldsanupperboundof(x2)+1327movesforsolutionstoRubik'scube.Wesimilarlyshowedthatnoneoftheelementsinanyofthe17cosetsatlevel16requiredmorethan26moves,againusingCubeExplorer.Incombinationwiththeabove,thisdemonstratesanupperboundof26movesforsolutionstoRubik'scube.7.4FurtherBruteForcingOurcontinuingworkisusingtheefcientbruteforcingtech-niquesgiveninSection6tofurtherreducetheupperboundfrom26to25moves.Weplantoachievethisbybruteforcingallcosetsatsomeearlylevelbythreemoves.Ourcurrentexperimentsindicatethatthereexistelementsthatcannotbebruteforcedbythreemovesouttolevel8.Directlyconsideringallcasesatlevel9iscomputationallyexpensive,asthereareover80millionlevel9cosets,eachwith3398unprovenelements(correspondingtothelastthreelevelsofthesquaresub-group).Instead,weusethenewbruteforcingtechniquesforcosetsatearlierlevels,and“project”theremainingunprovenelementstolaterlevels.Sofar,wehaveremovedover95%oftheelementsacrossallcosetsatlevel8.However,onlyabout10%ofthecosetsatlevel8havenoremainingcases,andrequireadditionalbruteforcing.Weanticipatethat,byusingourefcientbruteforcingtechnique,andsignicantcomputingpower,wewillbeablehandletheremainingcasesatlevels8and9,andthereforeprovethat25movessufceforRubik'scube.8.REFERENCES[1]GeneCooperman,LarryFinkelstein,andNamitaSarawagi.ApplicationsofCayleygraphs.InAAECC:AppliedAlgebra,AlgebraicAlgorithmsandError-CorrectingCodes,InternationalConference,pages367–378.LNCS,Springer-Verlag,1990.[2]AlexanderH.Frey,Jr.andDavidSingmaster.HandbookofCubikMath.EnslowPublishers,1982.[3]HerbertKociemba.CubeExplorer.http://kociemba.org/cube.htm,2006.[4]RichardKorf.FindingoptimalsolutionstoRubik'scubeusingpatterndatabases.InProceedingsoftheWorkshoponComputerGames(W31)atIJCAI-97,pages21–26,Nagoya,Japan,1997.[5]SilviuRadu.Rubikcanbesolvedin27f.http://cubezzz.homelinux.org/drupal/?q=,2006.[6]MichaelReid.Newupperbounds.http://www.math.rwth-aachen.de/˜Martin.Schoenert/Cube-Lovers/michael_re%id__new_upper_bounds.html,1995.[7]MichaelReid.Superiprequires20faceturns.http://www.math.rwth-aachen.de/˜Martin.Schoenert/Cube-Lovers/michael_re%id__superflip_requires_20_face_turns.html,1995.