C.R.WylieJr:

101PUZZLESINTHOUGHT&LOGIC

C.R.WYLIEJr.

DepartmentofMathematics,UniversityofUtah

PUZZLES

THOUGHTLOGIC

IN

AND

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INTRODUCTION

Althoughlifeisthegreatestpuzzleofall,thesepuzzlesarenottakenfromlife,andanyresem-blancetheymaybeartoactualpersonsorplacesisentirelycoincidental.

INTRODUCTION

Puzzlesofapurelylogicalnaturearedistinguishedfromriddles,ontheonehand,bythefactthattheyinvolvenoplayonwords,nodeliberatelydeceptivestatements,noguessing一inshort,no“catches”ofanykind.Theydifferfromquizzesandmostmathematicalpuzzles,ontheotherhand,inthatthoughtratherthanmemory,thatis,nativementalingenuityratherthanastoreofacquiredinformation,isthekeytotheirsolution.

Inorderthatthepuzzlesinthiscollectionshouldconformasnearlyaspossibletothisideal,everyefforthasbeenmadetokeepthefactualbasisofeachasmeageraspossible.Inaveryfewinstancestheuseofalittleelementaryalgebramaysimplifythesolution,butnoneactuallyrequiresanytechnicalinformationbeyondthemultiplicationtablesandthefactthat

distance=speed×time

Itisexpected,however,thatthereaderwillrecognizethatamanmustbeolderthanhischildren,thatwhentwopeoplewinamixeddoublesmatchoneismaleandtheotherisfe-male,andafewotherequallysimplefactsfromeverydayexperience.

Itisinterestingtoobservethatpuzzlesofthepurelylogicaltypeepitomizetheentirescientificprocess.Attheoutsetoneisconfrontedwithamassofmoreorlessunrelateddata.Fromthesefactsafewpositiveinferencescanperhapsbedrawnimmediately,butusuallyitisnecessarytosetuptentativeorworkinghypothesestoguidethesearchforasolution.Thevalidityofthesehypothesesmustthenbecare-fullycheckedbytestingtheirconsequencesforconsistency

Introduction

withtheoriginaldata.Ifinconsistenciesappear,thetenta-tiveassumptionsmustberejectedandotherssubstituteduntilfinallyaconsistentsetofconclusionsemerges.Thesecon-clusionsmustthenbetestedforuniquenesstodeterminewhethertheyalonemeettheconditionsoftheproblemorwhetherthereareothersequallyacceptable.

Thusbyrepetitionsofthefundamentalprocessofsettingupanhypothesis,drawingconclusionsfromit,andexaminingtheirconsistencywithinthetotalframeworkoftheproblem,theanswerisultimatelywrestedfromtheseeminglyinco-herentinformationinitiallyprovided.Andsoitisinscience,too.

Itisinherentinthenatureoflogicalpuzzlesthattheirsolutioncannotbereducedtoafixedpattern.Neverthelessitmaybehelpfulatthispointtooffersomegeneralsuggestionsonhowtoattackpuzzlesofthissort.Considerfirstthefol-lowingexample:

Boronoff,Pavlow,Revitsky,andSukarekarefourtalentedcreativeartists,oneadancer,oneapainter,oneasinger,andoneawriter(thoughnotnecessarilyrespectively).

(1)BoronoffandRevitskywereintheaudiencethenightthesingermadehisdebutontheconcertstage.

(2)BothPavlowandthewriterhavesatforportraitsbythepainter.

(3)Thewriter,whosebiographyofSukarekwasabest-seller,isplanningtowriteabiographyofBoronoff.

(4)BoronoffhasneverheardofRevitsky.

Whatiseachman'sartisticfield?

Tokeeptrackmentallyofthismanyfactsandthehypothe-sesandconclusionsbaseduponthemisconfusinganddiffi-cult.Inallbutthesimplestpuzzlesitisfarbettertoreducetheanalysissystematicallytoaseriesofwrittenmemoranda.

Introduction

Onemethodofaccomplishingthisistosetupanarrayinwhichallpossibilitiesareencompassed,thus:

BoronoffPavlow

RevitskySukarek

dancerpaintersingerwriter

Nowifweconsider,forexample,thatPavlowcannotbethedancerwewillplaceanX,say,oppositehisnameinthecolumnheadeddancer.OrifwedecidethatBoronoffmustbethepainterwewillplaceadifferentmark,sayanO,oppositehisnameinthiscolumn,whereuponwecanfilltheremainingsquaresinthisrowandcolumnwithX's(sincethereisonlyoneBoronoffandonlyonepainter).ClearlythesolutionwillbecompletewhenwesucceedinplacingconsistentlyexactlyoneOineachrowandineachcolumn,therebyshowingjustwhateachmanis.

Inthepresentproblemweknowfrom(1)thatneitherBoronoffnorRevitskyisthesinger,henceweplaceX'soppo-sitetheirnamesintheappropriatecolumn.From(2)weknowthatPavlowisneitherthepainternorthewriter,andfrom(3)weseethatthewriterisneitherBoronoffnorSukarek.WiththecorrespondingX'sdulyentered,thearraylookslikethis:

BoronoffPavlow

RevitskySukarek

dancerpaintersingerwriter

X

X

X

X

X

X

Introduction

ByeliminationitisnowclearthatRevitskyisthewriter.

HenceweenteranOoppositehisnameinthecolumnheadedwriterandfilltheremainingsquaresinhisrowwithX's.Moreover,accordingto(2),Revitskyhassatforthepainter,whileaccordingto(4)BoronoffdoesnotknowRevitsky.HenceBoronoffisnotthepainter,andsobyeliminationhemustbethedancer.ButthenneitherPavlownorSukarekcanbethedancer,andthisobservationleavessingerastheonlycategorypossibleforPavlow.Finally,Sukarekmustbethepainter,andthesolutioniscomplete.

Theprocedurewehavejustillustratedisalsoconvenientinidentification-puzzleswherethenecessaryinformationisgivenintheformofconditionalorcontingentstatements.Hereisasimpleexample:

(1)IfAisP,CisnotR.

(2)IfBisPorR,AisQ.

(3)IfAisQorR,BisP.

Determinethecorrespondencebetweenthesymbols(A,B,C)andthesymbols(P,Q,R).

Supposewebeginbyacceptingthehypothesis“AisP”.WethenconstructanarrayinwhichanOappearsoppositeAinthecolumnheadedPandX'sappearintheothersquaresintheA-rowandtheP-column:

AB

C

PQR

0

X

X

X

X

Nowfrom(1),CcannotbeR.HenceitmustbeQ,andneces-sarilythenBmustbeR.Butfrom(2),ifBisR,AmustbeQ,whichcontradictsboththeassumptionthatAisPandtheconclusionthatCisQ.ThisinconsistencyforcesustoabandonthehypothesisthatAisP.

Introduction

WecontinuenowbyconstructinganewarraybaseduponthefactthatAisnotP.Accordingto(3),ifAisnotPthenBmustbeP.HenceanOcanbeenteredoppositeBintheP-column,andtheremainingsquaresinthisrowandcolumncanbefilledwithX's:

AB

C

PQR

X

0

X

X

X

Thenfrom(2)weconcludethatAisQ,whereuponitfollows

thatCisR,andthesolutioniscomplete.

Insomepuzzlesthegiveninformationconsistsofasetofstatements,acertainnumberofwhichareknowntobefalsewithouttheuntrueassertionsbeingidentified.Puzzlesofthissortcanalsobeconvenientlyhandledthroughtheuseofarrays.Asanillustrationconsiderthefollowingexample:

ShortyFinelliwasfoundshottodeathonemorning,andthepolicewithbetterthanaverageluckhadthreered-hotsuspectsbehindbarsbynightfall.Thateven-ingthemenwerequestionedandmadethefollowingstatements.

Buck:(1)Ididn'tdoit.

(2)IneversawJoeybefore.

(3)Sure,IknewShorty.

Joey:(1)Ididn'tdoit

(2)BuckandTippyarebothpalsofmine.

(3)Buckneverkilledanybody.

Tippy:(1)Ididn'tdoit.

(2)Buckliedwhenhesaidhe'dneverseenJoeybefore.

(3)Idon'tknowwhodidit.

Ifoneandonlyoneofeachman'sstatementsisfalse,andifoneofthethreemenisactuallyguilty,whoisthemurderer?

Introduction

Heretheappropriatearrayisthefollowing

BuckJoeyTippy

123

andourproblemistoenteroneF(forfalse)andtwoT's(fortrue)ineachrowinamannerconsistentwiththegivenstatements.

AttheoutsetwecandrawthepositiveinferencethatTippyisinnocent.Forifhecommittedthecrime,thenhisfirstandthirdstatementsarebothfalse,contrarytothegivencondi-tionthatonlyoneofeachman'sassertionsisuntrue.ThisconclusioncannowberecordedasaToppositeTippyinthefirstcolumn.

Wearenowleftwithtwoalternatives:either(a)Buckistheguiltyone,or(b)Joeyistheguiltyone.Ifweassume(a),thenBuck'sfirststatementisfalseandJoey'slaststate-mentisfalse.UndertheconditionsoftheproblemthismeansthatBuck'ssecondandJoey'ssecondstatementmustbothbetrue.Butthisimpossiblesincetheyareclearlycontradictory.HencewemustabandontheassumptionthatBuckisthemurderer.ItfollowsthereforethatJoeyistheonewhokilledShorty,andthiscanbecheckedbyexaminingthecompletedarrayforthealternative(b):

BuckJoeyTippy

123

T

F

T

F

T

T

T

T

F

Introduction

Puzzlesconstructedbythecodingorsuppressionofdigitsinanarithmeticalcalculationrequirenomorethanattentiontoobviousnumericalfacts.Hereasinpuzzlesofthefore-going,moreverbaltypesitisalsohelpfultokeeptrackofcluesandconclusionsinanorderly,tabularway.Toillustrate,letusconsiderthefollowingexample:

Inacertainmultiplicationproblemeachdigitfrom

0to9wasreplacedbyadifferentletter,yieldingthecodedcalculation

Forwhatnumberdoeseachletterstand?

Tosystematizeourworkwefirstwriteinarowthediffer-entlettersappearingintheproblem:

ALERUMWINP

Overeachletterwewillwriteitsnumericalequivalentwhenwediscoverit.Inthecolumnsunderthevariousletterswewillrecordcluesandtentativehypotheses,beingcarefultoputallrelatedinferencesonthesamehorizontalline.

Inproblemsofthissortthedigits0and1canoftenbefound,oratleastrestrictedtoaveryfewpossibilities,bysimpleinspection.Forinstance,0canneveroccurastheleft-mostdigitofaninteger,andwhenanynumberismultipliedbyzerotheresultconsistsexclusivelyofzeros.Moreoverwhenanynumberismultipliedby1theresultisthatnumberitself.Inthepresentproblem,however,wecanidentify0by

Introduction

anevensimplerobservation.Forinthesecondcolumnfromtheright,NplusLequalsN,withnothingcarriedoverfromthecolumnontheright.HenceLmustbezero.

Inoursearchfor1wecaneliminateR,U,andMatonce,sincenoneofthese,asmultipliersinthesecondrow,repro-ducesALE.MoreoverEcannotbe1sinceUtimesEdoesnotyieldaproductendinginU.Atpresent,however,wehavenofurthercluesastowhether1isA,I,N,P,orW.

NowthepartialproductWUWLendsinL,whichweknowtobe0.HenceoneofthetwolettersUandEmustbe5.Lookingattheunitsdigitsoftheotherpartialproducts,weseethatbothM×EandR×EarenumbersendinginE.Amoment'sreflection(oraglanceatamultiplicationtable)showsthatEmustthereforebe5.

ButifEis5,thenbothRandMmustbeodd,sinceanevennumbersmultipliedby5wouldyieldaproductendingin0,whichisnotthecaseineitherthefirstorthirdpartialproduct.Moreover,bysimilarreasoningitisclearthatUisanevennumber.

AtthispointitisconvenienttoreturntoourarrayandlistunderUthevariouspossibilities,namely2,4,6,and8.OppositeeachofthesewerecordthecorrespondingvalueofWasreadfromthepartialproductWUWL,whoselasttwodigitsarenowdeterminedsincethefactorALEisknowntobe—05.ThesevaluesofWareeasilyseentobe1,2,3,and4.

FromaninspectionofthesecondcolumnfromtheleftwecannowdeducethecorrespondingpossibilitiesforR.Aswehavealreadynoted,Rmustbeodd;henceitsvalueistwiceWplus1(the1beingnecessarilycarriedoverfromthecolumnontheright).ThepossiblevaluesforRarethen3,5,7,and9,andourarraylookslikethis:

Introduction

05

ALERUMWINP

NowinthethirdcolumnfromtheleftintheexamplethesumofthedigitsW,U,andWmustbemorethan9,since1hadtobecarriedoverfromthiscolumnintothecolumnontheleft.Thevaluesinthefirsttworowsofthearrayaretoolowforthis,however,hencewecancrossoutbothoftheselines.

AfurtherconsiderationofthesumofthedigitsW,U,andWinthethirdcolumnfromtheleft,coupledwiththefactthatMisknowntobeodd,showsthatinthethirdrowofthearrayMmustbe3whileinthefourthrowitmustbe7.Thispermitsustorejectthethirdrowofthearrayalso,foritcontains3forbothMandW,whichisimpossible.Thecorrectsolutionmustthereforebetheonecontainedinthefourthrow.HenceRis9,Uis8,Mis7,andWis4.SubstitutingtheseintotheproblemitisasimplemattertodeterminethatAis6,Iis2,Nis3,andPis1.Thiscompletesthesolution.

Asanexampleofapuzzleinvolvingthesuppressionratherthanthecodingofdigits,considerthefollowing:

Inacertainprobleminlongdivisioneverydigitexcept7wassuppressed,yielding

Restorethemissingdigits.

Introduction

Theobviouspointofattackhereisthefirstpartialproduct,—77,sinceitisthemostnearlydeterminednumberintheproblem.Now,theonlyone-digitnumberswhoseproductendsin7are3and9.Hencethefirstdigitinthequotientmustbeoneofthesenumbersandthelastdigitinthedivisormustbetheother.Ifweconsiderthepossibledivisorsoftheform—9andmultiplyeachby3,wefindthattheonlyonewhichyieldsaproductoftheform—77is59whichgives177.Alternatively,ifwetrydivisorsoftheform—3andmultiplyeachby9wefindthatonly53yieldsaproductoftheform—77.Wemustrejectthefirstofthesetwopossibili-ties,however,sincewhen59ismultipliedbytheseconddigitinthequotient,namely7,theresultis413,whereasaccord-ingtotheproblemthesecondpartialproductisoftheform—7—.Thisleaves53astheuniquepossibilityforthedivisorand9asthefirstdigitofthequotient.Finallyweobservethatthelastdigitofthequotientmustbe1sincethelastpartialproductcontainsjusttwodigits.Knowingthatthedivisoris59andthequotientis971,wecanmultiplythesenumberstoobtainthedividend.Therestoftheproblemcanthenbereconstructedatonce.

Mostofthepuzzlesinthiscollectionhaveuniquesolutions.

Afewleadtoseveraldifferentsolutions,acircumstanceal-waysindicatedinthestatementoftheproblem.Therearealsoafewpuzzlesinwhichtheobjectisnottofindananswerbuttoprovethatthereisnone,thatistoshowthatthegivendata,takenalltogether,areincompatible.Asanillustrationofapuzzleofthistype,considerthefollowingcodedsub-traction:

Ifeachletterissupposedtostandforadifferentdigit,provethatthereisnopossiblewaytoassignauniquedigittoeachlettertoformacorrectsubtraction.

WenoticefirstthatintheleftmostcolumnthesubtractionofTfromEleaves0.HenceEmustbeexactly1morethan

Introduction

T(the1havingbeenborrowedfromEforuseinthesecondcolumn).Nowintherightmostcolumn,TminusEyieldsE.(SinceEisgreaterthanT,1hadtobeborrowedfromthecolumnonthelefttomakethissubtractionpossible.)Ortoputitinthereversesense,EplusEisatwo-digitnumberhavingTintheunitsplace.HenceTmustbeeven,andofcoursedifferentfrom0sinceitappearsastheleftmostdigitinthesecondrowoftheproblem.Wethereforehavethefollowingpossibilities:

T:2468

E:6789

Amongthesethereisonlyonepair,namelyE=9,T=8,whichmeetsthefurtherrequirementthatEis1morethanT.

Nowconsiderthesubtractioninthesecondcolumnfromtheright.Wehavealreadyobservedthat1hadtobeborrowedfromtheHforuseinthecolumnontheright.HenceE,thatis9,takenawayfrom1lessthanHleavesV.Butfirstborrowing1fromanumberandthentaking9awayfromwhatremainsisclearlyjustthesameastaking10awayfromtheoriginalnumber.Andwhen10issubtractedfromanynumber,theunitsdigitofthenumbernecessarilyappearsunchangedastheunitsdigitoftheanswer.HencetheresultofthesubtractioninthesecondcolumnfromtherightmustbeHandcannotbethedifferentdigitV.Thisinescapablecontradictionprovesthattheproblemcannotbedecodedtoproduceacorrectsubtraction.

Manyofthepuzzlesinthisbookareeasytosolve,othersareratherdifficult.Itislikely,however,thatonepersonwillfindsomeeasythatanotherwillfindhard,andviceversa,formethodsofanalysisdifferfromindividualtoindividual.Withinwidelimitsthetimerequiredtosolveaparticularproblemisoflittlesignificanceasanindicationofaperson'sabilitytoreason.Foronepersonmaybypurechanceselectthecorrectassumptionforhisfirsttrial,whileanequallyalertindividualmayunluckilyexploreanynumberoffruit-lesshypothesesbeforehereachestherightone.

Introduction

Thepuzzleshereinareallnewinsubstance,thoughnotinform,forapuzzleofanentirelynewformisalmostunimagin-able.Nonehasbeenpublishedelsewhere.Allhavebeencare-fullycheckedandeach,whateveritsothermeritsorfaults,hasbeenformulatedsoastobesolvablebylogicalreasoningwithonlythebarestminimumofacquiredinformation.

Andnow—pleasantpuzzling!

C.R.WYLIEJR.

SaltLakeCity,Utah

PUZZLES

Solutionsinbackofthebook.

Inacertainbankthepositionsofcashier,manager,andtellerareheldbyBrown,JonesandSmith,thoughnotneces-sarilyrespectively.

Theteller,whowasanonlychild,earnstheleast.

Smith,whomarriedBrown'ssister,earnsmorethanthemanager.

Whatpositiondoeseachmanfill?

Clark,DawandFullermaketheirlivingascarpenter,painterandplumber,thoughnotnecessarliyrespectively.

Thepainterrecentlytriedtogetthecarpentertodo

someworkforhim,butwastoldthatthecarpenterwas

outdoingsomeremodelingfortheplumber.

Theplumbermakesmoremoneythanthepainter.DawmakesmoremoneythanClark.

FullerhasneverheardofDaw.

Whatiseachman'soccupation?

Dorothy,Jean,Virginia,Bill,Jim,andTomaresixyoungpersonswhohavebeenclosefriendsfromtheirchildhood.Theywentthroughhighschoolandcollegetogether,andwhentheyfinallypairedoffandbecameengagednothingwoulddobutatripleannouncementparty.Naturallytheywantedtobreakthenewstotheirfriendsinanunusualfashion,andaftersomethoughttheydecideduponthisscheme.

Atjusttherightmomentduringthepartyeveryonewasgivenacardbearingthecrypticinformation:

Whonowaresixwillsoonbethree,

Andgailyweconfessit,

Buthowwe'vechosenyoumayknow

Nosoonerthanyouguessit.

Tom,whoisolderthanJim,isDorothy'sbrother.Virginiaistheoldestgirl.

Thetotalageofeachcouple-to-beisthesameal-thoughnotwoofusarethesameage.

JimandJeanaretogetherasoldasBillandDorothy.

Whatthreeengagementswereannouncedattheparty?

Mr.Carter,Mr.Flynn,Mr.Milne,andMr.SavageservethelittletownofMilfordasarchitect,banker,druggist,andgrocer,thoughnotnecessarilyrespectively.Eachman'sincomeisawholenumberofdollars.

Thedruggistearnsexactlytwiceasmuchasthegrocer,thearchitectearnsexactlytwiceasmuchasthedruggist,andthebankerearnsexactlytwiceasmuchasthearchitect.

AlthoughMr.CarterdoesnotmakemoremoneythanMr.Flynn,Mr.FlynndoesnotmaketwiceasmuchasMr.Carter.

Mr.Savageearnsexactly$3776morethanMr.Milne.

Whatiseachman'soccupation?

Brown,Clark,Jones,andSmitharethenamesofthemenwhohold,thoughnotnecessarilyrespectively,thepositionsofaccountant,cashier,manager,andpresidentintheFirstNa-tionalBankofFairport.

Althoughthecashierbeatshimconsistently,thepresidentwillplaychesswithnooneelseinthebank.

Boththemanagerandthecashierarebetterchess

playersthantheaccountant.

JonesandSmitharenextdoorneighborsandfre-quentlyplaychesstogetherintheevening.

ClarkplaysabettergameofchessthanJones.

Theaccountantlivesnearthepresidentbutnotnearanyoftheothers.

Whatpositiondoeseachmanhold?

Clark,Jones,Morgan,andSmitharefourmenwhoseoccu-

pationarebutcher,druggist,grocer,andpoliceman,thoughnotnecessarilyrespectively.

ClarkandJonesareneighborsandtaketurnsdrivingeachothertowork.

JonesmakesmoremoneythanMorgan.

ClarkbeatsSmithregularlyatbowling.

Thebutcheralwayswalkstowork.

Thepolicemandoesnotnotlivenearthedruggist.

Theonlytimethegrocerandthepolicemanevermet

waswhenthepolicemanarrestedthegrocerforspeed-ing.

Thepolicemanmakesmoremoneythanthedruggist

orthegrocer.

Whatiseachman'soccupation?

Brown,Clark,JonesandSmitharefoursubstantialcitizenswhoservetheircommunityasarchitect,banker,doctor,andlawyer,thoughnotnecessarilyrespectively.

Brown,whoismoreconservativethanJonesbutmoreliberalthanSmith,isabettergolferthanthemenwhoareolderthanheisandhasalargerincomethanthemenwhoareyoungerthanClark.

Thebanker,whoearnsmorethanthearchitect,isneithertheyoungestnortheoldest.

Thedoctor,whoisapoorergolferthanthelawyer,islessconservativethanthearchitect.

Asmightbeexpected,theoldestmanisthemostconservativeandhasthelargestincome,andtheyoung-estmanisthebestgolfer.

Whatiseachman'sprofession?

Inacertaindepartmentstorethepositionofbuyer,cashier,clerk,floorwalker,andmanagerareheld,thoughnotneces-sarilyrespectively,byMissAmes,MissBrown,Mr.Conroy,Mr.Davis,andMr.Evans.

Thecashierandthemanagerwereroommatesin

college.

Thebuyerisabachelor.

EvansandMissAmeshavehadonlybusinesscon-tactswitheachother.

Mrs.Conroywasgreatlydisappointedwhenherhus-bandtoldherthatthemanagerhadrefusedtogivehimaraise.

Davisisgoingtobethebestmanwhentheclerkandthecashieraremarried.

Whatpositiondoeseachpersonhold?

Thepositionsofbuyer,cashier,clerk,floorwalker,andman-agerintheEmpireDepartmentStoreareheldbyMessrs.Allen,Bennett,Clark,Davis,andEwing.

Thecashierandthefloorwalkereatlunchfrom11:30to12:30,theotherseatfrom12:30to1:30.

Mrs.AllenandMrs.Clarkaresisters.

AllenandBennettalwaysbringtheirlunchandplaycribbageduringtheirlunchhour.

DavisandEwinghavenothingtodowitheachothersincethedayDavis,returningfromlunchearlierthanusual,foundEwingalreadygoneandreportedhimtothemanager.

Thecashierandtheclerksharebachelorquarters.

Whatpositiondoeseachmanfll?

Jane,Janice,Jack,Jasper,andJimarethenamesoffivehighschoolchums.TheirlastnamesinoneorderoranotherareCarter,Carver,Clark,Clayton,andCramer.

Jasper'smotherisdead.

Indeferencetoacertainverywealthyaunt,Mr.