HowtoSolveIt

ANewAspectofMathematicalMethod

G.POLYAWithanewforewordbyJohnH.Conway

Copyright©1945byPrincetonUniversityPressCopyright©renewed1973byPrincetonUniversityPress

SecondEditionCopyright©1957byG.PolyaSecondEditionCopyright©renewed1985

byPrincetonUniversityPressAllRightsReserved

FirstPrincetonPaperbackprinting,1971Secondprinting,1973

FirstPrincetonScienceLibraryEdition,1988

ExpandedPrincetonScienceLibraryEdition,withanewforewordbyJohnH.Conway,2004

LibraryofCongressControlNumber2004100613

ISBN-13:978-0-691-11966-3(pbk.)

ISBN-10:0-691-11966-X(pbk.)

BritishLibraryCataloging-in-PublicationDataisavailable

Printedonacid-freepaper.∞

psl.princeton.edu

PrintedintheUnitedStatesofAmerica

357910864

FromthePrefacetotheFirstPrinting

A great discovery solves a great problem but there is a grain of discovery in the solution of anyproblem. Your problem may be modest; but if it challenges your curiosity and brings into play yourinventivefaculties,andifyousolveitbyyourownmeans,youmayexperiencethetensionandenjoythetriumphofdiscovery.Suchexperiencesatasusceptibleagemaycreateatasteformentalworkandleavetheirimprintonmindandcharacterforalifetime.

Thus, a teacherofmathematicshas agreat opportunity. If he fills his allotted timewithdrillinghisstudentsinroutineoperationshekillstheirinterest,hamperstheirintellectualdevelopment,andmisuseshisopportunity.Butifhechallengesthecuriosityofhisstudentsbysettingthemproblemsproportionatetotheirknowledge,andhelpsthemtosolvetheirproblemswithstimulatingquestions,hemaygivethematastefor,andsomemeansof,independentthinking.

Alsoastudentwhosecollegecurriculumincludessomemathematicshasasingularopportunity.Thisopportunity is lost,ofcourse, ifheregardsmathematicsasasubject inwhichhehas toearnsoandsomuchcreditandwhichheshouldforgetafterthefinalexaminationasquicklyaspossible.Theopportunitymaybe losteven if the studenthas somenatural talent formathematicsbecausehe,aseverybodyelse,must discover his talents and tastes; he cannot know that he likes raspberry pie if he has never tastedraspberrypie.Hemaymanagetofindout,however,thatamathematicsproblemmaybeasmuchfunasacrosswordpuzzle,orthatvigorousmentalworkmaybeanexerciseasdesirableasafastgameoftennis.Havingtastedthepleasureinmathematicshewillnotforgetiteasilyandthenthereisagoodchancethatmathematicswillbecomesomethingforhim:ahobby,oratoolofhisprofession,orhisprofession,oragreatambition.

Theauthorremembersthetimewhenhewasastudenthimself,asomewhatambitiousstudent,eagertounderstand a little mathematics and physics. He listened to lectures, read books, tried to take in thesolutions and facts presented, but there was a question that disturbed him again and again: “Yes, thesolutionseemstowork,itappearstobecorrect;buthowisitpossibletoinventsuchasolution?Yes,thisexperimentseemstowork,thisappearstobeafact;buthowcanpeoplediscoversuchfacts?Andhowcould I invent or discover such things by myself?” Today the author is teaching mathematics in auniversity;hethinksorhopesthatsomeofhismoreeagerstudentsasksimilarquestionsandhetriestosatisfy their curiosity. Trying to understand not only the solution of this or that problem but also themotivesandproceduresofthesolution,andtryingtoexplainthesemotivesandprocedurestoothers,hewasfinallyledtowritethepresentbook.Hehopesthatitwillbeusefultoteacherswhowishtodeveloptheirstudents’abilitytosolveproblems,andtostudentswhoarekeenondevelopingtheirownabilities.

Although the present book pays special attention to the requirements of students and teachers ofmathematics,itshouldinterestanybodyconcernedwiththewaysandmeansofinventionanddiscovery.Suchinterestmaybemorewidespreadthanonewouldassumewithoutreflection.Thespacedevotedbypopularnewspapers andmagazines to crosswordpuzzles andother riddles seems to show that peoplespend some time in solvingunpractical problems.Behind the desire to solve this or that problem thatconfers nomaterial advantage, there may be a deeper curiosity, a desire to understand the ways andmeans,themotivesandprocedures,ofsolution.

Thefollowingpagesarewrittensomewhatconcisely,butassimplyaspossible,andarebasedonalongandseriousstudyofmethodsofsolution.Thissortofstudy,calledheuristicbysomewriters,isnotinfashionnowadaysbuthasalongpastand,perhaps,somefuture.

Studyingthemethodsofsolvingproblems,weperceiveanotherfaceofmathematics.Yes,mathematicshastwofaces;itistherigorousscienceofEuclidbutitisalsosomethingelse.MathematicspresentedintheEuclideanwayappearsasasystematic,deductivescience;butmathematicsinthemakingappearsas

anexperimental,inductivescience.Bothaspectsareasoldasthescienceofmathematicsitself.Butthesecondaspect isnewinonerespect;mathematics“instatunascendi,” in theprocessofbeinginvented,hasneverbeforebeenpresented inquite thismanner to thestudent,or to the teacherhimself,or to thegeneralpublic.

The subject of heuristic has manifold connections; mathematicians, logicians, psychologists,educationalists,evenphilosophersmayclaimvariouspartsof it asbelonging to their specialdomains.Theauthor,wellawareofthepossibilityofcriticismfromoppositequartersandkeenlyconsciousofhislimitations, has one claim to make: he has some experience in solving problems and in teachingmathematicsonvariouslevels.

Thesubject ismore fullydealtwith inamoreextensivebookby theauthorwhich ison theway tocompletion.

StanfordUniversity,August1,1944

FromthePrefacetotheSeventhPrinting

IamgladtosaythatIhavenowsucceededinfulfilling,atleastinpart,apromisegivenintheprefacetothefirstprinting:ThetwovolumesInductionandAnalogyinMathematicsandPatternsofPlausibleInferencewhichconstitutemyrecentworkMathematicsandPlausibleReasoning continue the lineofthinkingbeguninHowtoSolveIt.

Zurich,August30,1954

PrefacetotheSecondEdition

Thepresentsecondeditionadds,besidesa fewminor improvements,anewfourthpart,“Problems,Hints,Solutions.”

As this edition was being prepared for print, a study appeared (Educational Testing Service,Princeton,N.J.;cf.Time,June18,1956)whichseemstohaveformulatedafewpertinentobservations—theyarenotnewtothepeopleintheknow,butitwashightimetoformulatethemforthegeneralpublic—:“...mathematicshasthedubioushonorofbeingtheleastpopularsubjectinthecurriculum...Futureteachers pass through the elementary schools learning to detest mathematics . . . They return to theelementaryschooltoteachanewgenerationtodetestit.”

I hope that thepresent edition,designed forwiderdiffusion,will convince someof its readers thatmathematics,besidesbeinganecessaryavenuetoengineeringjobsandscientificknowledge,maybefunandmayalsoopenupavistaofmentalactivityonthehighestlevel.

Zurich,June30,1956

Contents

FromthePrefacetotheFirstPrintingFromthePrefacetotheSeventhPrintingPrefacetotheSecondEdition“HowtoSolveIt”listForewordIntroduction

PARTI.INTHECLASSROOM

Purpose

1.Helpingthestudent2.Questions,recommendations,mentaloperations3.Generality4.Commonsense5.Teacherandstudent.Imitationandpractice

Maindivisions,mainquestions

6.Fourphases7.Understandingtheproblem8.Example9.Devisingaplan10.Example11.Carryingouttheplan12.Example13.Lookingback14.Example15.Variousapproaches16.Theteacher’smethodofquestioning17.Goodquestionsandbadquestions

Moreexamples

18.Aproblemofconstruction19.Aproblemtoprove

20.Arateproblem

PARTII.HOWTOSOLVEITAdialogue

PARTIII.SHORTDICTIONARYOFHEURISTICAnalogyAuxiliaryelementsAuxiliaryproblemBolzanoBrightideaCanyouchecktheresult?Canyouderivetheresultdifferently?Canyouusetheresult?CarryingoutConditionContradictory†CorollaryCouldyouderivesomethingusefulfromthedata?Couldyourestatetheproblem?†DecomposingandrecombiningDefinitionDescartesDetermination,hope,successDiagnosisDidyouuseallthedata?Doyouknowarelatedproblem?Drawafigure†ExamineyourguessFiguresGeneralizationHaveyouseenitbefore?HereisaproblemrelatedtoyoursandsolvedbeforeHeuristic

HeuristicreasoningIfyoucannotsolvetheproposedproblemInductionandmathematicalinductionInventor’sparadoxIsitpossibletosatisfythecondition?LeibnitzLemmaLookattheunknownModernheuristicNotationPappusPedantryandmasteryPracticalproblemsProblemstofind,problemstoproveProgressandachievementPuzzlesReductioadabsurdumandindirectproofRedundant†RoutineproblemRulesofdiscoveryRulesofstyleRulesofteachingSeparatethevariouspartsoftheconditionSettingupequationsSignsofprogressSpecializationSubconsciousworkSymmetryTerms,oldandnewTestbydimensionThefuturemathematicianTheintelligentproblem-solverTheintelligentreaderThetraditionalmathematicsprofessorVariationoftheproblem

Whatistheunknown?Whyproofs?WisdomofproverbsWorkingbackwards

PARTIV.PROBLEMS,HINTS,SOLUTIONSProblemsHintsSolutions

†Containsonlycross-references.

HOWTOSOLVEIT

UNDERSTANDINGTHEPROBLEM

First.

Youhavetounderstandtheproblem.What is the unknown?What are the data?What is the condition? Is it possible to satisfy the condition? Is the condition sufficient todeterminetheunknown?Orisitinsufficient?Orredundant?Orcontradictory?Drawafigure.Introducesuitablenotation.Separatethevariouspartsofthecondition.Canyouwritethemdown?

DEVISINGAPLAN

Second.

Findtheconnectionbetweenthedataandtheunknown.Youmaybeobligedtoconsiderauxiliaryproblemsifanimmediateconnectioncannotbefound.

Youshouldobtaineventuallyaplanofthesolution.Haveyouseenitbefore?Orhaveyouseenthesameprobleminaslightlydifferentform?Doyouknowarelatedproblem?Doyouknowatheoremthatcouldbeuseful?Lookattheunknown!Andtrytothinkofafamiliarproblemhavingthesameorasimilarunknown.Hereisaproblemrelatedtoyoursandsolvedbefore.Couldyouuseit?Couldyouuseitsresult?Couldyouuseitsmethod?Shouldyouintroducesomeauxiliaryelementinordertomakeitsusepossible?Couldyourestatetheproblem?Couldyourestateitstilldifferently?Gobacktodefinitions.Ifyoucannotsolve theproposedproblemtry tosolvefirstsomerelatedproblem.Couldyouimagineamoreaccessiblerelatedproblem?Amoregeneral problem?Amore special problem?An analogous problem?Couldyou solve a part of the problem?Keeponly a part of thecondition,droptheotherpart;howfaris theunknownthendetermined,howcanitvary?Couldyouderivesomethingusefulfromthedata?Couldyouthinkofotherdataappropriatetodeterminetheunknown?Couldyouchangetheunknownorthedata,orbothifnecessary,sothatthenewunknownandthenewdataarenearertoeachother?Didyouuseallthedata?Didyouusethewholecondition?Haveyoutakenintoaccountallessentialnotionsinvolvedintheproblem?

CARRYINGOUTTHEPLAN

Third.

Carryoutyourplan.Carryingoutyourplanofthesolution,checkeachstep.Canyouseeclearlythatthestepiscorrect?Canyouprovethatitiscorrect?

LOOKINGBACK

Fourth.

Examinethesolutionobtained.Canyouchecktheresult?Canyouchecktheargument?Canyouderivetheresultdifferently?Canyouseeitataglance?Canyouusetheresult,orthemethod,forsomeotherproblem?

ForewordbyJohnH.Conway

HowtoSolveItisawonderfulbook!ThisIrealizedwhenIfirstreadrightthroughitasastudentmanyyearsago,butithastakenmealongtimetoappreciatejusthowwonderfulitis.Whyisthat?Onepartoftheansweristhatthebookisunique.Inallmyyearsasastudentandteacher,IhaveneverseenanotherthatlivesuptoGeorgePolya’stitlebyteachingyouhowtogoaboutsolvingproblems.A.H.Schoenfeldcorrectly described its importance in his 1987 article “Polya, Problem Solving, and Education” inMathematicsMagazine:“Formathematicseducationandtheworldofproblemsolvingitmarkedalineofdemarcationbetweentwoeras,problemsolvingbeforeandafterPolya.”

Itisoneofthemostsuccessfulmathematicsbookseverwritten,havingsoldoveramillioncopiesandbeen translated into seventeen languages since it first appeared in 1945. Polya later wrote twomorebooks about the art of doing mathematics, Mathematics and Plausible Reasoning (1954) andMathematicalDiscovery(twovolumes,1962and1965).

Thebook’stitlemakesitseemthatitisdirectedonlytowardstudents,butinfactitisaddressedjustasmuch to their teachers. Indeed,asPolyaremarks inhis introduction, thefirstpartof thebook takes theteacher’sviewpointmoreoftenthanthestudent’s.

Everybody gains that way. The student who reads the book on his own will find that overhearingPolya’scommentstohisnon-existentteachercanbringthatdesirablepersonintobeing,asanimaginarybutveryhelpful figure leaningoverone’sshoulder.This iswhathappened tome,andnaturallyImadeheavyuseoftheremarksI’dfoundmostimportantwhenImyselfstartedteachingafewyearslater.

ButitwassometimebeforeIreadthebookagain,andwhenIdid,IsuddenlyrealizedthatitwasevenmorevaluablethanI’dthought!ManyofPolya’sremarksthathadn’thelpedmeasastudentnowmademeabetterteacherofthosewhoseproblemshaddifferedfrommine.PolyahadmetmanymorestudentsthanIhad,andhadobviouslythoughtveryhardabouthowtobesthelpallofthemlearnmathematics.Perhapshismostimportantpointisthatlearningmustbeactive.Ashesaidinalectureonteaching,“Mathematics,yousee, isnota spectator sport.Tounderstandmathematicsmeans tobeable todomathematics.Andwhatdoesitmean[tobe]doingmathematics?Inthefirstplace,itmeanstobeabletosolvemathematicalproblems.”

Itisoftensaidthattoteachanysubjectwell,onehastounderstandit“atleastaswellasone’sstudentsdo.”Itisaparadoxicaltruththattoteachmathematicswell,onemustalsoknowhowtomisunderstanditatleasttotheextentone’sstudentsdo!Ifateacher’sstatementcanbeparsedintwoormoreways,itgoeswithoutsayingthatsomestudentswillunderstanditonewayandothersanother,withresultsthatcanvaryfrom the hilarious to the tragic. J. E. Littlewood gives two amusing examples of assumptions that caneasilybemadeunconsciouslyandmisleadingly.First,he remarks that thedescriptionof thecoordinateaxes(“OxandOyasin2dimensions,Ozvertical”)inLamb’sbookMechanicsisincorrectforhim,sincehealwaysworkedinanarmchairwithhisfeetup!Then,afteraskinghowhisreaderwouldpresentthepictureofaclosedcurvelyingallononesideofitstangent,hestatesthattherearefourmainschools(toleftorrightofverticaltangent,oraboveorbelowhorizontalone)andthatbylecturingwithoutafigure,presumingthatthecurvewastotherightofitsverticaltangent,hehadunwittinglymadenonsensefortheotherthreeschools.

I know of no better remedy for such presumptions than Polya’s counsel: before trying to solve aproblem, the student should demonstrate his or her understanding of its statement, preferably to a realteacher,but in lieuof that, toanimaginedone.Experiencedmathematiciansknowthatoftenthehardestpart of researching a problem is understanding precisely what that problem says. They often follow

Polya’swiseadvice:“Ifyoucan’tsolveaproblem,thenthereisaneasierproblemyoucan’tsolve:findit.”

Readerswholearnfromthisbookwillalsowanttolearnaboutitsauthor’slife.1GeorgePolyawasbornGyörgyPólya(hedroppedtheaccentssometimelater)onDecember13,1887,

inBudapest,Hungary, toJakabPólyaandhiswife, theformerAnnaDeutsch.HewasbaptizedintotheRomanCatholicfaith,towhichJakab,Anna,andtheirthreepreviouschildren,Jenő,Ilona,andFlóra,hadconvertedfromJudaisminthepreviousyear.Theirfifthchild,László,wasbornfouryearslater.

JakabhadchangedhissurnamefromPolláktothemoreHungarian-soundingPólyafiveyearsbeforeGyörgywasborn,believingthatthismighthelphimobtainauniversitypost,whichheeventuallydid,butonlyshortlybeforehisuntimelydeathin1897.

At the Dániel Berzsenyi Gymnasium, György studied Greek, Latin, and German, in addition toHungarian.Itissurprisingtolearnthattherehewasseeminglyuninterestedinmathematics,hisworkingeometry deemed merely “satisfactory” compared with his “outstanding” performance in literature,geography,andothersubjects.Hisfavoritesubject,outsideofliterature,wasbiology.

He enrolled at the University of Budapest in 1905, initially studying law, which he soon droppedbecausehefoundittooboring.HethenobtainedthecertificationneededtoteachLatinandHungarianatagymnasium,acertificationthatheneverusedbutofwhichheremainedproud.Eventuallyhisprofessor,BernátAlexander,advisedhim that tohelphisstudies inphilosophy,heshould takesomemathematicsandphysicscourses.Thiswashowhecametomathematics.Later,hejokedthathe“wasn’tgoodenoughforphysics,andwastoogoodforphilosophy—mathematicsisinbetween.”

InBudapesthewastaughtphysicsbyEötvösandmathematicsbyFejérandwasawardedadoctorateafter spending the academic year 1910–11 in Vienna, where he took some courses byWirtinger andMertens.HespentmuchofthenexttwoyearsinGöttingen,wherehemetmanymoremathematicians—Klein,Caratheodory,Hilbert,Runge,Landau,Weyl,Hecke,Courant,andToeplitz—andin1914visitedParis,wherehebecameacquaintedwithPicardandHadamardandlearnedthatHurwitzhadarrangedanappointmentforhiminZürich.Heacceptedthisposition,writinglater:“IwenttoZürichinordertobenearHurwitz,andwewereinclosetouchforaboutsixyears,frommyarrivalinZürichin1914tohispassing[in1919].Iwasverymuchimpressedbyhimandeditedhisworks.”

Ofcourse,theFirstWorldWartookplaceduringthisperiod.ItinitiallyhadlittleeffectonPolya,whohadbeendeclaredunfitforserviceintheHungarianarmyastheresultofasoccerwound.Butlaterwhenthearmy,moredesperatelyneedingrecruits,demanded thathereturn to fight forhiscountry,hisstrongpacifistviewsledhimtorefuse.Asaconsequence,hewasunabletovisitHungaryformanyyears,andinfactdidnotdosountil1967,fifty-fouryearsafterheleft.

Inthemeantime,hehadtakenSwisscitizenshipandmarriedaSwissgirl,StellaVeraWeber,in1918.Between1918and1919,hepublishedpapersonawiderangeofmathematicalsubjects,suchasseries,numbertheory,combinatorics,votingsystems,astronomy,andprobability.Hewasmadeanextraordinaryprofessor at theZürichETH in1920, and a fewyears later he andGáborSzegő published their bookAufgabenundLehrsatzeausderAnalysis (“ProblemsandTheoremsinAnalysis”),describedbyG.L.AlexandersonandL.H.LangeintheirobituaryofPolyaas“amathematicalmasterpiecethatassuredtheirreputations.”

Thatbookappearedin1925,afterPolyahadobtainedaRockefellerFellowshiptoworkinEngland,where he collaborated with Hardy and Littlewood on what later became their book Inequalities(Cambridge University Press, 1936). He used a second Rockefeller Fellowship to visit PrincetonUniversity in 1933, and while in the United States was invited by H. F. Blichfeldt to visit StanfordUniversity,whichhegreatlyenjoyed,andwhichultimatelybecamehishome.PolyaheldaprofessorshipatStanfordfrom1943untilhisretirementin1953,anditwasthere,in1978,thathetaughthislastcourse,

incombinatorics;hediedonSeptember7,1985,attheageofninety-seven.SomereaderswillwanttoknowaboutPolya’smanycontributionstomathematics.Mostofthemrelate

toanalysisandaretootechnicaltobeunderstoodbynon-experts,butafewareworthmentioning.Inprobabilitytheory,Polyaisresponsibleforthenow-standardterm“CentralLimitTheorem”andfor

provingthattheFouriertransformofaprobabilitymeasureisacharacteristicfunctionandthatarandomwalkontheintegerlatticecloseswithprobability1ifandonlyifthedimensionisatmost2.

In geometry, Polya independently re-enumerated the seventeen plane crystallographic groups (theirfirstenumeration,byE.S.Fedorov,havingbeenforgotten)andtogetherwithP.Nigglidevisedanotationforthem.

In combinatorics, Polya’s Enumeration Theorem is now a standard way of counting configurationsaccording to their symmetry. It has been described by R. C. Read as “a remarkable theorem in aremarkablepaper,andalandmarkinthehistoryofcombinatorialanalysis.”HowtoSolveItwaswritteninGermanduringPolya’stimeinZürich,whichendedin1940,whenthe

Europeansituationforcedhimto leavefor theUnitedStates.Despite thebook’seventualsuccess, fourpublishersrejectedtheEnglishversionbeforePrincetonUniversityPressbroughtitoutin1945.Intheirhands,HowtoSolveIt rapidlybecame—andcontinuestobe—oneof themostsuccessfulmathematicalbooksofalltime.

1The following biographical information is taken from that given by J. J. O’Connor and E. F. Robertson in the MacTutor History ofMathematicsArchive(www-gap.dcs.st-and.ac.uk/~history/).

http://www-gap.dcs.st-and.ac.uk/~history/

Introduction

The following considerations are grouped around the preceding list of questions and suggestionsentitled“HowtoSolveIt.”Anyquestionorsuggestionquotedfromitwillbeprintedinitalics,andthewholelistwillbereferredtosimplyas“thelist”oras“ourlist.”

Thefollowingpageswilldiscussthepurposeofthelist,illustrateitspracticalusebyexamples,andexplaintheunderlyingnotionsandmentaloperations.Bywayofpreliminaryexplanation,thismuchmaybesaid:If,usingthemproperly,youaddressthesequestionsandsuggestionstoyourself, theymayhelpyoutosolveyourproblem.If,usingthemproperly,youaddressthesamequestionsandsuggestionstooneofyourstudents,youmayhelphimtosolvehisproblem.

Thebookisdividedintofourparts.Thetitleofthefirstpartis“IntheClassroom.”Itcontainstwentysections.Eachsectionwillbequoted

byitsnumberinheavytypeas,forinstance,“section7.”Sections1to5discussthe“Purpose”ofourlistingeneralterms.Sections6to17explainwhatarethe“MainDivisions,MainQuestions”ofthelist,anddiscussafirstpracticalexample.Sections18,19,20add“MoreExamples.”

The title of the very short second part is “How to Solve It.” It iswritten in dialogue; a somewhatidealizedteacheranswersshortquestionsofasomewhatidealizedstudent.

The third andmost extensive part is a “Short Dictionary of Heuristic”; we shall refer to it as the“Dictionary.” It contains sixty-seven articles arranged alphabetically. For example, themeaning of thetermHEURISTIC(setinsmallcapitals)isexplainedinanarticlewiththistitleonpage112.Whenthetitleofsuchanarticleisreferredtowithinthetextitwillbesetinsmallcapitals.Certainparagraphsofafewarticles are more technical; they are enclosed in square brackets. Some articles are fairly closelyconnectedwith the firstpart towhich theyadd further illustrationsandmore specificcomments.Otherarticlesgosomewhatbeyond theaimof thefirstpartofwhich theyexplain thebackground.There isakey-articleonMODERNHEURISTIC.ItexplainstheconnectionofthemainarticlesandtheplanunderlyingtheDictionary;itcontainsalsodirectionshowtofindinformationaboutparticularitemsofthelist.Itmustbeemphasizedthatthereisacommonplanandacertainunity,becausethearticlesoftheDictionaryshowthegreatestoutwardvariety.Thereareafewlongerarticlesdevotedtothesystematicthoughcondenseddiscussionofsomegeneral theme;otherscontainmorespecificcomments,stillotherscross-references,orhistoricaldata,orquotations,oraphorisms,orevenjokes.

TheDictionaryshouldnotbereadtooquickly;itstextisoftencondensed,andnowandthensomewhatsubtle.ThereadermayrefertotheDictionaryforinformationaboutparticularpoints.Ifthesepointscomefromhisexperiencewithhisownproblemsorhisownstudents,thereadinghasamuchbetterchancetobeprofitable.

The titleof the fourthpart is“Problems,Hints,Solutions.” Itproposesa fewproblems to themoreambitiousreader.Eachproblemisfollowed(inproperdistance)bya“hint”thatmayrevealawaytotheresultwhichisexplainedinthe“solution.”

Wehavementioned repeatedly the“student” and the“teacher” andweshall refer to themagainandagain.Itmaybegoodtoobservethatthe“student”maybeahighschoolstudent,oracollegestudent,oranyoneelsewhoisstudyingmathematics.Alsothe“teacher”maybeahighschoolteacher,oracollegeinstructor,oranyoneinterestedinthetechniqueofteachingmathematics.Theauthorlooksatthesituationsometimesfromthepointofviewofthestudentandsometimesfromthatoftheteacher(thelattercaseispreponderantinthefirstpart).Yetmostofthetime(especiallyinthethirdpart)thepointofviewisthatofapersonwhoisneitherteachernorstudentbutanxioustosolvetheproblembeforehim.

HowtoSolveIt

PARTI.INTHECLASSROOM

PURPOSE1.Helpingthestudent.Oneofthemostimportanttasksoftheteacheristohelphisstudents.Thistask

isnotquiteeasy;itdemandstime,practice,devotion,andsoundprinciples.Thestudentshouldacquireasmuchexperienceofindependentworkaspossible.Butifheisleftalone

withhisproblemwithoutanyhelporwithinsufficienthelp,hemaymakenoprogressatall.Iftheteacherhelpstoomuch,nothingislefttothestudent.Theteachershouldhelp,butnottoomuchandnottoolittle,sothatthestudentshallhaveareasonableshareofthework.

Ifthestudentisnotabletodomuch,theteachershouldleavehimatleastsomeillusionofindependentwork.Inordertodoso,theteachershouldhelpthestudentdiscreetly,unobtrusively.

The best is, however, to help the student naturally. The teacher should put himself in the student’splace,heshouldseethestudent’scase,heshouldtrytounderstandwhatisgoingoninthestudent’smind,andaskaquestionorindicateastepthatcouldhaveoccurredtothestudenthimself.

2. Questions, recommendations, mental operations. Trying to help the student effectively butunobtrusivelyandnaturally, the teacher is led toask thesamequestionsand to indicate thesamestepsagainandagain.Thus,incountlessproblems,wehavetoaskthequestion:Whatistheunknown?Wemayvarythewords,andaskthesamethinginmanydifferentways:Whatisrequired?Whatdoyouwanttofind?Whatareyousupposedtoseek?Theaimofthesequestionsistofocusthestudent’sattentionuponthe unknown. Sometimes, we obtain the same effect more naturally with a suggestion: Look at theunknown!Questionandsuggestionaimatthesameeffect;theytendtoprovokethesamementaloperation.

Itseemedtotheauthorthatitmightbeworthwhiletocollectandtogroupquestionsandsuggestionswhicharetypicallyhelpfulindiscussingproblemswithstudents.Thelistwestudycontainsquestionsandsuggestionsofthissort,carefullychosenandarranged;theyareequallyusefultotheproblem-solverwhoworksbyhimself.Ifthereaderissufficientlyacquaintedwiththelistandcansee,behindthesuggestion,the action suggested, he may realize that the list enumerates, indirectly,mental operations typicallyusefulforthesolutionofproblems.Theseoperationsarelistedintheorderinwhichtheyaremostlikelytooccur.

3.Generalityisanimportantcharacteristicofthequestionsandsuggestionscontainedinourlist.Takethequestions:What is theunknown?Whatare thedata?What is thecondition? These questions aregenerallyapplicable,wecanaskthemwithgoodeffectdealingwithallsortsofproblems.Theiruseisnot restricted to any subject-matter. Our problem may be algebraic or geometric, mathematical ornonmathematical,theoreticalorpractical,aseriousproblemoramerepuzzle;itmakesnodifference,thequestionsmakesenseandmighthelpustosolvetheproblem.

Thereisarestriction, infact,but ithasnothingtodowiththesubject-matter.Certainquestionsandsuggestionsofthelistareapplicableto“problemstofind”only,notto“problemstoprove.”Ifwehaveaproblemofthelatterkindwemustusedifferentquestions;seePROBLEMSTOFIND,PROBLEMSTOPROVE.

4. Common sense. The questions and suggestions of our list are general, but, except for theirgenerality,theyarenatural,simple,obvious,andproceedfromplaincommonsense.Takethesuggestion:Lookat theunknown!Andtry to thinkofa familiarproblemhavingthesameorasimilarunknown.Thissuggestionadvisesyoutodowhatyouwoulddoanyhow,withoutanyadvice,ifyouwereseriouslyconcernedwithyourproblem.Areyouhungry?Youwishtoobtainfoodandyouthinkoffamiliarwaysofobtainingfood.Haveyouaproblemofgeometricconstruction?Youwishtoconstructatriangleandyouthinkof familiarwaysofconstructinga triangle.Haveyouaproblemofanykind?Youwish to findacertainunknown,andyouthinkoffamiliarwaysoffindingsuchanunknown,orsomesimilarunknown.Ifyoudosoyoufollowexactlythesuggestionwequotedfromourlist.Andyouareontherighttrack,too;thesuggestionisagoodone,itsuggeststoyouaprocedurewhichisveryfrequentlysuccessful.

Allthequestionsandsuggestionsofourlistarenatural,simple,obvious,justplaincommonsense;buttheystateplaincommonsenseingeneralterms.Theysuggestacertainconductwhichcomesnaturallytoanypersonwhoisseriouslyconcernedwithhisproblemandhassomecommonsense.Butthepersonwhobehaves the rightway usually does not care to express his behavior in clearwords and, possibly, hecannotexpressitso;ourlisttriestoexpressitso.

5.Teacherandstudent.Imitationandpractice.Therearetwoaimswhichtheteachermayhaveinviewwhenaddressingtohisstudentsaquestionorasuggestionof the list:First, tohelp thestudent tosolvetheproblemathand.Second,todevelopthestudent’sabilitysothathemaysolvefutureproblemsbyhimself.

Experience shows that thequestions and suggestionsofour list, appropriatelyused,very frequentlyhelpthestudent.Theyhavetwocommoncharacteristics,commonsenseandgenerality.Astheyproceedfromplaincommonsensetheyveryoftencomenaturally;theycouldhaveoccurredtothestudenthimself.Astheyaregeneral,theyhelpunobtrusively;theyjustindicateageneraldirectionandleaveplentyforthestudenttodo.

But the twoaimswementionedbeforearecloselyconnected; if the student succeeds in solving theproblemat hand, he adds a little to his ability to solve problems.Then,we shouldnot forget that ourquestionsaregeneral, applicable inmanycases. If the samequestion is repeatedlyhelpful, the studentwillscarcelyfailtonoticeitandhewillbeinducedtoaskthequestionbyhimselfinasimilarsituation.Asking the question repeatedly, hemay succeedonce in eliciting the right idea.By such a success, hediscoverstherightwayofusingthequestion,andthenhehasreallyassimilatedit.

Thestudentmayabsorbafewquestionsofourlistsowellthatheisfinallyabletoputtohimselftheright question in the right moment and to perform the corresponding mental operation naturally andvigorously.Suchastudenthascertainlyderived thegreatestpossibleprofit fromour list.Whatcan theteacherdoinordertoobtainthisbestpossibleresult?

Solving problems is a practical skill like, let us say, swimming.We acquire any practical skill byimitationandpractice.Tryingtoswim,youimitatewhatotherpeopledowiththeirhandsandfeettokeeptheir heads above water, and, finally, you learn to swim by practicing swimming. Trying to solveproblems,youhavetoobserveandtoimitatewhatotherpeopledowhensolvingproblemsand,finally,youlearntodoproblemsbydoingthem.

Theteacherwhowishestodevelophisstudents’abilitytodoproblemsmustinstillsomeinterestforproblems into theirmindsandgive themplentyofopportunity for imitationandpractice. If the teacherwishestodevelopinhisstudentsthementaloperationswhichcorrespondtothequestionsandsuggestionsof our list, he puts these questions and suggestions to the students as often as he can do so naturally.Moreover,whentheteachersolvesaproblembeforetheclass,heshoulddramatizehisideasalittleandhe should put to himself the same questionswhich he useswhen helping the students. Thanks to suchguidance,thestudentwilleventuallydiscovertherightuseofthesequestionsandsuggestions,anddoingsohewillacquiresomething that ismore important than theknowledgeofanyparticularmathematical

fact.

MAINDIVISIONS,MAINQUESTIONS6.Fourphases.Tryingtofindthesolution,wemayrepeatedlychangeourpointofview,ourwayof

lookingattheproblem.Wehavetoshiftourpositionagainandagain.Ourconceptionoftheproblemislikelytoberatherincompletewhenwestartthework;ouroutlookisdifferentwhenwehavemadesomeprogress;itisagaindifferentwhenwehavealmostobtainedthesolution.

In order to group conveniently the questions and suggestions of our list, we shall distinguish fourphasesofthework.First,wehavetounderstandtheproblem;wehavetoseeclearlywhatisrequired.Second,wehavetoseehowthevariousitemsareconnected,howtheunknownislinkedtothedata,inordertoobtaintheideaofthesolution,tomakeaplan.Third,wecarryoutourplan.Fourth,we lookbackatthecompletedsolution,wereviewanddiscussit.

Eachofthesephaseshasitsimportance.Itmayhappenthatastudenthitsuponanexceptionallybrightidea and jumping all preparations blurts out with the solution. Such lucky ideas, of course, are mostdesirable,butsomethingveryundesirableandunfortunatemayresultifthestudentleavesoutanyofthefourphaseswithouthavingagoodidea.Theworstmayhappenifthestudentembarksuponcomputationsor constructions without having understood the problem. It is generally useless to carry out detailswithouthavingseenthemainconnection,orhavingmadeasortofplan.Manymistakescanbeavoidedif,carryingouthisplan,thestudentcheckseachstep.Someofthebesteffectsmaybelostifthestudentfailstoreexamineandtoreconsiderthecompletedsolution.

7.Understandingtheproblem.Itisfoolishtoansweraquestionthatyoudonotunderstand.Itissadtoworkforanendthatyoudonotdesire.Suchfoolishandsadthingsoftenhappen,inandoutofschool,buttheteachershouldtrytopreventthemfromhappeninginhisclass.Thestudentshouldunderstandtheproblem.Butheshouldnotonlyunderstandit,heshouldalsodesireitssolution.Ifthestudentislackingin understanding or in interest, it is not always his fault; the problem should bewell chosen, not toodifficult and not too easy, natural and interesting, and some time should be allowed for natural andinterestingpresentation.

Firstofall,theverbalstatementoftheproblemmustbeunderstood.Theteachercancheckthis,uptoacertain extent; he asks the student to repeat the statement, and the student should be able to state theproblem fluently. The student should also be able to point out the principal parts of the problem, theunknown,thedata,thecondition.Hence,theteachercanseldomaffordtomissthequestions:Whatistheunknown?Whatarethedata?Whatisthecondition?

Thestudentshouldconsidertheprincipalpartsoftheproblemattentively,repeatedly,andfromvarioussides. If there is a figure connectedwith the problemhe shoulddrawa figure and point out on it theunknown and the data. If it is necessary to give names to these objects he should introduce suitablenotation;devotingsomeattentiontotheappropriatechoiceofsigns,heisobligedtoconsidertheobjectsforwhichthesignshavetobechosen.Thereisanotherquestionwhichmaybeusefulinthispreparatorystage provided thatwe do not expect a definitive answer but just a provisional answer, a guess: Is itpossibletosatisfythecondition?

(In the exposition of Part II [p. 33] “Understanding the problem” is subdivided into two stages:“Gettingacquainted”and“Workingforbetterunderstanding.”)

8. Example. Let us illustrate some of the points explained in the foregoing section. We take thefollowingsimpleproblem:Findthediagonalofarectangularparallelepipedofwhichthelength,thewidth,andtheheightareknown.

In order to discuss this problem profitably, the students must be familiar with the theorem ofPythagoras,andwithsomeofitsapplicationsinplanegeometry,buttheymayhaveverylittlesystematicknowledge in solid geometry.The teachermay rely here upon the student’s unsophisticated familiaritywithspatialrelations.

The teachercanmake theprobleminterestingbymaking itconcrete.Theclassroomisa rectangularparallelepipedwhosedimensionscouldbemeasured,andcanbeestimated;thestudentshavetofind,to“measureindirectly,”thediagonaloftheclassroom.Theteacherpointsoutthelength,thewidth,andtheheight of the classroom, indicates the diagonal with a gesture, and enlivens his figure, drawn on theblackboard,byreferringrepeatedlytotheclassroom.

Thedialoguebetweentheteacherandthestudentsmaystartasfollows:“Whatistheunknown?”“Thelengthofthediagonalofaparallelepiped.”“Whatarethedata?”“Thelength,thewidth,andtheheightoftheparallelepiped.”“Introducesuitablenotation.Whichlettershoulddenotetheunknown?”“x.”“Whichletterswouldyouchooseforthelength,thewidth,andtheheight?”“a,b,c.”“Whatisthecondition,linkinga,b,c,andx?”“xisthediagonaloftheparallelepipedofwhicha,b,andcarethelength,thewidth,andtheheight.”“Isitareasonableproblem?Imean,istheconditionsufficienttodeterminetheunknown?”“Yes, it is. Ifweknowa,b,c,weknowtheparallelepiped. If theparallelepiped isdetermined, the

diagonalisdetermined.”9.Devisingaplan.Wehaveaplanwhenweknow,orknowat least inoutline,whichcalculations,

computations, or constructions we have to perform in order to obtain the unknown. The way fromunderstandingtheproblemtoconceivingaplanmaybelongandtortuous.Infact,themainachievementinthe solutionof aproblem is to conceive the ideaof aplan.This ideamayemergegradually.Or, afterapparentlyunsuccessful trialsandaperiodofhesitation, itmayoccursuddenly, inaflash,asa“brightidea.”Thebestthattheteachercandoforthestudentistoprocureforhim,byunobtrusivehelp,abrightidea.Thequestionsandsuggestionswearegoingtodiscusstendtoprovokesuchanidea.

Inordertobeabletoseethestudent’sposition,theteachershouldthinkofhisownexperience,ofhisdifficultiesandsuccessesinsolvingproblems.

Weknow,ofcourse,thatitishardtohaveagoodideaifwehavelittleknowledgeofthesubject,andimpossibletohaveitifwehavenoknowledge.Goodideasarebasedonpastexperienceandformerlyacquiredknowledge.Mererememberingisnotenoughforagoodidea,butwecannothaveanygoodideawithoutrecollectingsomepertinentfacts;materialsalonearenotenoughforconstructingahousebutwecannotconstructahousewithoutcollectingthenecessarymaterials.Thematerialsnecessaryforsolvingamathematical problemare certain relevant itemsof our formerly acquiredmathematical knowledge, asformerlysolvedproblems,or formerlyproved theorems.Thus, it isoftenappropriate tostart theworkwiththequestion:Doyouknowarelatedproblem?

Thedifficultyisthatthereareusuallytoomanyproblemswhicharesomewhatrelatedtoourpresentproblem,thatis,havesomepointincommonwithit.Howcanwechoosetheone,orthefew,whicharereally useful? There is a suggestion that puts our finger on an essential common point: Look at theunknown!Andtrytothinkofafamiliarproblemhavingthesameorasimilarunknown.

Ifwesucceedinrecallingaformerlysolvedproblemwhichiscloselyrelatedtoourpresentproblem,wearelucky.Weshouldtrytodeservesuchluck;wemaydeserveitbyexploitingit.Hereisaproblem

relatedtoyoursandsolvedbefore.Couldyouuseit?Theforegoingquestions,wellunderstoodandseriouslyconsidered,veryoftenhelp tostart theright

trainof ideas;but theycannothelpalways, theycannotworkmagic. If theydonotwork,wemust lookaroundforsomeotherappropriatepointofcontact,andexplorethevariousaspectsofourproblem;wehavetovary,totransform,tomodifytheproblem.Couldyourestatetheproblem?Someofthequestionsof our list hint specific means to vary the problem, as generalization, specialization, use of analogy,droppingapartof thecondition, and soon; thedetails are importantbutwecannotgo into themnow.Variation of the problem may lead to some appropriate auxiliary problem: If you cannot solve theproposedproblemtrytosolvefirstsomerelatedproblem.

Trying to apply various known problems or theorems, considering various modifications,experimentingwithvariousauxiliaryproblems,wemaystraysofarfromouroriginalproblemthatweareindangeroflosingitaltogether.Yetthereisagoodquestionthatmaybringusbacktoit:Didyouuseallthedata?Didyouusethewholecondition?

10. Example. We return to the example considered in section 8. As we left it, the students justsucceededinunderstandingtheproblemandshowedsomemildinterestinit.Theycouldnowhavesomeideasoftheirown,someinitiative.Iftheteacher,havingwatchedsharply,cannotdetectanysignofsuchinitiativehehastoresumecarefullyhisdialoguewiththestudents.Hemustbepreparedtorepeatwithsomemodification thequestionswhich the studentsdonot answer.Hemust beprepared tomeet oftenwiththedisconcertingsilenceofthestudents(whichwillbeindicatedbydots.....).“Doyouknowarelatedproblem?”.....“Lookattheunknown!Doyouknowaproblemhavingthesameunknown?”.....“Well,whatistheunknown?”“Thediagonalofaparallelepiped.”“Doyouknowanyproblemwiththesameunknown?”“No.Wehavenothadanyproblemyetaboutthediagonalofaparallelepiped.”“Doyouknowanyproblemwithasimilarunknown?”.....“You see, thediagonal is a segment, the segment of a straight line.Didyounever solve a problem

whoseunknownwasthelengthofaline?”“Ofcourse,wehavesolvedsuchproblems.Forinstance,tofindasideofarighttriangle.”“Good!Hereisaproblemrelatedtoyoursandsolvedbefore.Couldyouuseit?”.....“Youwereluckyenoughtorememberaproblemwhichisrelatedtoyourpresentoneandwhichyou

solvedbefore.Wouldyouliketouseit?Couldyouintroducesomeauxiliaryelementinordertomakeitsusepossible?”

FIG.1

.....“Lookhere,theproblemyourememberedisaboutatriangle.Haveyouanytriangleinyourfigure?”Let us hope that the last hint was explicit enough to provoke the idea of the solution which is to

introducearighttriangle,(emphasizedinFig.1)ofwhichtherequireddiagonalisthehypotenuse.Yettheteachershouldbepreparedforthecasethateventhisfairlyexplicithintisinsufficienttoshakethetorporofthestudents;andsoheshouldbepreparedtouseawholegamutofmoreandmoreexplicithints.

“Wouldyouliketohaveatriangleinthefigure?”“Whatsortoftrianglewouldyouliketohaveinthefigure?”“Youcannotfindyetthediagonal;butyousaidthatyoucouldfindthesideofatriangle.Now,what

willyoudo?”“Couldyoufindthediagonal,ifitwereasideofatriangle?”When,eventually,withmoreor lesshelp, the students succeed in introducing thedecisiveauxiliary

element,therighttriangleemphasizedinFig.1,theteachershouldconvincehimselfthatthestudentsseesufficientlyfaraheadbeforeencouragingthemtogointoactualcalculations.

“I think that it was a good idea to draw that triangle. You have now a triangle; but have you theunknown?”

“Theunknownisthehypotenuseofthetriangle;wecancalculateitbythetheoremofPythagoras.”“Youcan,ifbothlegsareknown;butarethey?”“One leg is given, it is c. And the other, I think, is not difficult to find. Yes, the other leg is the

hypotenuseofanotherrighttriangle.”“Verygood!NowIseethatyouhaveaplan.”11.Carryingouttheplan.Todeviseaplan,toconceivetheideaofthesolutionisnoteasy.Ittakesso

muchtosucceed;formerlyacquiredknowledge,goodmentalhabits,concentrationuponthepurpose,andonemorething:goodluck.Tocarryouttheplanismucheasier;whatweneedismainlypatience.

Theplangivesageneraloutline;wehavetoconvinceourselvesthatthedetailsfitintotheoutline,andsowehavetoexaminethedetailsoneaftertheother,patiently,tilleverythingisperfectlyclear,andnoobscurecornerremainsinwhichanerrorcouldbehidden.

If the studenthas reallyconceivedaplan, the teacherhasnowa relativelypeaceful time.Themaindanger is that thestudentforgetshisplan.Thismayeasilyhappenif thestudentreceivedhisplanfromoutside,andaccepteditontheauthorityof theteacher;but ifheworkedfor ithimself,evenwithsomehelp,andconceivedthefinalideawithsatisfaction,hewillnotlosethisideaeasily.Yettheteachermustinsistthatthestudentshouldcheckeachstep.

We may convince ourselves of the correctness of a step in our reasoning either “intuitively” or

“formally.”Wemayconcentrateuponthepointinquestiontillweseeitsoclearlyanddistinctlythatwehavenodoubtthatthestepiscorrect;orwemayderivethepointinquestionaccordingtoformalrules.(Thedifferencebetween“insight”and“formalproof”isclearenoughinmanyimportantcases;wemayleavefurtherdiscussiontophilosophers.)

Themain point is that the student should be honestly convinced of the correctness of each step. Incertaincases, the teachermayemphasize thedifferencebetween“seeing” and“proving”:Canyou seeclearlythatthestepiscorrect?Butcanyoualsoprovethatthestepiscorrect?

12.Example.Letusresumeourworkatthepointwhereweleftitattheendofsection10.Thestudent,atlast,hasgottheideaofthesolution.Heseestherighttriangleofwhichtheunknownxisthehypotenuseandthegivenheightcisoneofthelegs;theotherlegisthediagonalofaface.Thestudentmust,possibly,beurgedtointroducesuitablenotation.Heshouldchoosey todenotethatotherleg, thediagonalof thefacewhosesidesareaandb.Thus,hemayseemoreclearlytheideaofthesolutionwhichistointroduceanauxiliaryproblemwhoseunknownisy.Finally,workingatoneright triangleafter theother,hemayobtain(seeFig.1)

x2=y2+c2y2=a2+b2

andhence,eliminatingtheauxiliaryunknowny,

The teacher has no reason to interrupt the student if he carries out these details correctly except,possibly,towarnhimthatheshouldcheckeachstep.Thus,theteachermayask:

“Canyouseeclearlythatthetrianglewithsidesx,y,cisarighttriangle?”To this question the student may answer honestly “Yes” but he could be much embarrassed if the

teacher,notsatisfiedwiththeintuitiveconvictionofthestudent,shouldgoonasking:“Butcanyouprovethatthistriangleisarighttriangle?”Thus,theteachershouldrathersuppressthisquestionunlesstheclasshashadagoodinitiationinsolid

geometry. Even in the latter case, there is some danger that the answer to an incidental questionmaybecomethemaindifficultyforthemajorityofthestudents.

13.Lookingback.Evenfairlygoodstudents,whentheyhaveobtainedthesolutionoftheproblemandwrittendownneatly theargument,shut theirbooksandlookforsomethingelse.Doingso, theymissanimportantandinstructivephaseofthework.Bylookingbackatthecompletedsolution,byreconsideringandreexaminingtheresultandthepaththatledtoit,theycouldconsolidatetheirknowledgeanddeveloptheirabilitytosolveproblems.Agoodteachershouldunderstandandimpressonhisstudentstheviewthatnoproblemwhateveriscompletelyexhausted.Thereremainsalwayssomethingtodo;withsufficientstudy and penetration, we could improve any solution, and, in any case, we can always improve ourunderstandingofthesolution.

Thestudenthasnowcarriedthroughhisplan.Hehaswrittendownthesolution,checkingeachstep.Thus,heshouldhavegoodreasonstobelievethathissolutioniscorrect.Nevertheless,errorsarealwayspossible,especiallyiftheargumentislongandinvolved.Hence,verificationsaredesirable.Especially,if there is some rapidand intuitiveprocedure to test either the resultor theargument, it shouldnotbeoverlooked.Canyouchecktheresult?Canyouchecktheargument?

Inordertoconvinceourselvesofthepresenceorofthequalityofanobject,weliketoseeandtotouchit.Andaswepreferperception through twodifferent senses, sowepreferconvictionby twodifferent

proofs:Canyouderivetheresultdifferently?Weprefer,ofcourse,ashortandintuitiveargument toalongandheavyone:Canyouseeitataglance?

One of the first and foremost duties of the teacher is not to give his students the impression thatmathematicalproblemshavelittleconnectionwitheachother,andnoconnectionatallwithanythingelse.We have a natural opportunity to investigate the connections of a problem when looking back at itssolution.Thestudentswillfindlookingbackatthesolutionreallyinterestingiftheyhavemadeanhonesteffort,andhavetheconsciousnessofhavingdonewell.Thentheyareeagertoseewhatelsetheycouldaccomplish with that effort, and how they could do equally well another time. The teacher shouldencouragethestudentstoimaginecasesinwhichtheycouldutilizeagaintheprocedureused,orapplytheresultobtained.Canyouusetheresult,orthemethod,forsomeotherproblem?

14. Example. In section 12, the students finally obtained the solution: If the three edges of arectangularparallelogram,issuedfromthesamecorner,area,b,c,thediagonalis

Can you check the result? The teacher cannot expect a good answer to this question frominexperiencedstudents.Thestudents,however,shouldacquirefairlyearlytheexperiencethatproblems“inletters”haveagreatadvantageoverpurelynumericalproblems;iftheproblemisgiven“inletters”itsresult is accessible to several tests to which a problem “in numbers” is not susceptible at all. Ourexample,althoughfairlysimple,issufficienttoshowthis.Theteachercanaskseveralquestionsabouttheresultwhichthestudentsmayreadilyanswerwith“Yes”;butananswer“No”wouldshowaseriousflawintheresult.“Didyouuseallthedata?Doallthedataa,b,cappearinyourformulaforthediagonal?”“Length,width,andheightplaythesameroleinourquestion;ourproblemissymmetricwithrespectto

a,b,c.Istheexpressionyouobtainedforthediagonalsymmetricina,b,c?Does it remainunchangedwhena,b,careinterchanged?”

“Our problem is a problem of solid geometry: to find the diagonal of a parallelepipedwith givendimensionsa,b,c.Ourproblemisanalogoustoaproblemofplanegeometry:tofindthediagonalofarectanglewithgivendimensionsa,b. Is theresultofour ‘solid’problemanalogous to theresultof the‘plane’problem?”

“Iftheheightcdecreases,andfinallyvanishes,theparallelepipedbecomesaparallelogram.Ifyouputc = 0 in your formula, do you obtain the correct formula for the diagonal of the rectangularparallelogram?”

“Iftheheightcincreases,thediagonalincreases.Doesyourformulashowthis?”“Ifallthreemeasuresa,b,coftheparallelepipedincreaseinthesameproportion,thediagonalalso

increasesinthesameproportion.If,inyourformula,yousubstitute12a,12b,12cfora,b,crespectively,theexpressionofthediagonal,owingtothissubstitution,shouldalsobemultipliedby12.Isthatso?”

“Ifa,b,caremeasuredinfeet,yourformulagivesthediagonalmeasuredinfeettoo;butifyouchangeallmeasuresintoinches,theformulashouldremaincorrect.Isthatso?”

(Thetwolastquestionsareessentiallyequivalent;seeTESTBYDIMENSION.)Thesequestionshaveseveralgoodeffects.First,anintelligentstudentcannothelpbeingimpressedby

the fact that the formula passes so many tests. He was convinced before that the formula is correctbecausehederiveditcarefully.Butnowheismoreconvinced,andhisgaininconfidencecomesfromadifferentsource;itisduetoasortof“experimentalevidence.”Then,thankstotheforegoingquestions,thedetailsof the formulaacquirenewsignificance, andare linkedupwithvarious facts.The formulahastherefore a better chance of being remembered, the knowledge of the student is consolidated. Finally,these questions can be easily transferred to similar problems. After some experience with similar

problems, an intelligent student may perceive the underlying general ideas: use of all relevant data,variationofthedata,symmetry,analogy.Ifhegetsintothehabitofdirectinghisattentiontosuchpoints,hisabilitytosolveproblemsmaydefinitelyprofit.Canyouchecktheargument?Torechecktheargumentstepbystepmaybenecessaryindifficultand

importantcases.Usually, it isenoughtopickout“touchy”pointsforrechecking.Inourcase, itmaybeadvisabletodiscussretrospectivelythequestionwhichwaslessadvisabletodiscussasthesolutionwasnot yet attained:Canyouprove that the trianglewith sides x, y, c is a right triangle? (See the end ofsection12.)Canyouusetheresultorthemethodforsomeotherproblem?Withalittleencouragement,andafter

one or two examples, the students easily find applications which consist essentially in giving someconcreteinterpretation to theabstractmathematicalelementsof theproblem.The teacherhimselfusedsuch a concrete interpretation as he took the room in which the discussion takes place for theparallelepipedoftheproblem.Adullstudentmaypropose,asapplication,tocalculatethediagonalofthecafeteria instead of the diagonal of the classroom. If the students do not volunteer more imaginativeremarks,theteacherhimselfmayputaslightlydifferentproblem,forinstance:“Beinggiventhelength,thewidth, and the height of a rectangular parallelepiped, find the distance of the center from one of thecorners.”

Thestudentsmayusetheresultoftheproblemtheyjustsolved,observingthatthedistancerequiredisone half of the diagonal they just calculated. Or they may use themethod, introducing suitable righttriangles(thelatteralternativeislessobviousandsomewhatmoreclumsyinthepresentcase).

After this application, the teacher may discuss the configuration of the four diagonals of theparallelepiped,andthesixpyramidsofwhichthesixfacesarethebases,thecenterthecommonvertex,and the semidiagonals the lateral edges.When the geometric imagination of the students is sufficientlyenlivened,theteachershouldcomebacktohisquestion:Canyouusetheresult,orthemethod,forsomeotherproblem?Nowthere isabetterchance that thestudentsmayfindsomemore interestingconcreteinterpretation,forinstance,thefollowing:

“In thecenterof the flat rectangular topofabuildingwhich is21yards longand16yardswide, aflagpoleistobeerected,8yardshigh.Tosupportthepole,weneedfourequalcables.Thecablesshouldstartfromthesamepoint,2yardsunderthetopofthepole,andendatthefourcornersofthetopofthebuilding.Howlongiseachcable?”

Thestudentsmayusethemethodoftheproblemtheysolvedindetailintroducingarighttriangleinaverticalplane,andanotheroneinahorizontalplane.Ortheymayusetheresult,imaginingarectangularparallelepipedofwhichthediagonal,x,isoneofthefourcablesandtheedgesare

a=10.5b=8c=6.

Bystraightforwardapplicationoftheformula,x=14.5.Formoreexamples,seeCANYOUUSETHERESULT?15.Variousapproaches.Letusstillretain,forawhile,theproblemweconsideredintheforegoing

sections8, 10, 12, 14.Themainwork, thediscoveryof theplan,wasdescribed in section10.Let usobservethattheteachercouldhaveproceededdifferently.Startingfromthesamepointasinsection10,hecouldhavefollowedasomewhatdifferentline,askingthefollowingquestions:“Doyouknowanyrelatedproblem?”“Doyouknowananalogousproblem?”“You see, the proposed problem is a problem of solid geometry. Could you think of a simpler

analogousproblemofplanegeometry?”“You see, the proposed problem is about a figure in space, it is concernedwith the diagonal of a

rectangularparallelepiped.Whatmightbeananalogousproblemaboutafigureintheplane?Itshouldbeconcernedwith—thediagonal—of—arectangular—”

“Parallelogram.”Thestudents,evenif theyareveryslowandindifferent,andwerenotabletoguessanythingbefore,

areobligedfinallytocontributeatleastaminutepartoftheidea.Besides,ifthestudentsaresoslow,theteachershouldnottakeupthepresentproblemabouttheparallelepipedwithouthavingdiscussedbefore,inordertopreparethestudents,theanalogousproblemabouttheparallelogram.Then,hecangoonnowasfollows:“Hereisaproblemrelatedtoyoursandsolvedbefore.Canyouuseit?”“Shouldyouintroducesomeauxiliaryelementinordertomakeitsusepossible?”Eventually, the teacher may succeed in suggesting to the students the desirable idea. It consists in

conceiving the diagonal of the given parallelepiped as the diagonal of a suitable parallelogramwhichmustbeintroducedintothefigure(asintersectionoftheparallelepipedwithaplanepassingthroughtwooppositeedges).Theideaisessentiallythesameasbefore(section10)buttheapproachisdifferent.Insection10,thecontactwiththeavailableknowledgeofthestudentswasestablishedthroughtheunknown;a formerlysolvedproblemwas recollectedbecause itsunknownwas thesameas thatof theproposedproblem.Inthepresentsectionanalogyprovidesthecontactwiththeideaofthesolution.

16. The teacher’s method of questioning shown in the foregoing sections 8, 10, 12, 14, 15 isessentially this:Beginwith a general question or suggestion of our list, and, if necessary, comedowngradually to more specific and concrete questions or suggestions till you reach one which elicits aresponse in thestudent’smind. Ifyouhave tohelp the studentexploithis idea, startagain, ifpossible,fromageneralquestionorsuggestioncontainedinthelist,andreturnagaintosomemorespecialoneifnecessary;andsoon.

Ofcourse,our list is justa first listof thiskind; itseems tobesufficient for themajorityofsimplecases,butthereisnodoubtthatitcouldbeperfected.Itisimportant,however,thatthesuggestionsfromwhichwestartshouldbesimple,natural,andgeneral,andthattheirlistshouldbeshort.

Thesuggestionsmustbesimpleandnaturalbecauseotherwisetheycannotbeunobtrusive.The suggestionsmust be general, applicable not only to the present problembut to problemsof all

sorts,iftheyaretohelpdeveloptheabilityofthestudentandnotjustaspecialtechnique.The list must be short in order that the questions may be often repeated, unartificially, and under

varyingcircumstances;thus,thereisachancethattheywillbeeventuallyassimilatedbythestudentandwillcontributetothedevelopmentofamentalhabit.

Itisnecessarytocomedowngraduallytospecificsuggestions,inorderthatthestudentmayhaveasgreatashareoftheworkaspossible.

This method of questioning is not a rigid one; fortunately so, because, in these matters, any rigid,mechanical,pedanticalprocedureisnecessarilybad.Ourmethodadmitsacertainelasticityandvariation,itadmitsvariousapproaches(section15),itcanbeandshouldbesoappliedthatquestionsaskedbytheteachercouldhaveoccurredtothestudenthimself.

If a readerwishes to try themethodhere proposed in his class he should, of course, proceedwithcaution.He should study carefully the example introduced in section8, and the following examples insections18,19,20.Heshouldpreparecarefullytheexampleswhichheintendstodiscuss,consideringalsovariousapproaches.Heshouldstartwithafewtrialsandfindoutgraduallyhowhecanmanagethemethod,howthestudentstakeit,andhowmuchtimeittakes.

17. Good questions and bad questions. If the method of questioning formulated in the foregoingsectioniswellunderstoodithelpstojudge,bycomparison,thequalityofcertainsuggestionswhichmaybeofferedwiththeintentionofhelpingthestudents.

Letusgobacktothesituationasitpresenteditselfat thebeginningofsection10whenthequestionwasasked:Doyouknowarelatedproblem?Insteadofthis,withthebestintentiontohelpthestudents,thequestionmaybeoffered:CouldyouapplythetheoremofPythagoras?

Theintentionmaybethebest,butthequestionisabouttheworst.Wemustrealizeinwhatsituationitwasoffered;thenweshallseethatthereisalongsequenceofobjectionsagainstthatsortof“help.”

(1)Ifthestudentisneartothesolution,hemayunderstandthesuggestionimpliedbythequestion;butif he is not, he quite possiblywill not see at all the point at which the question is driving. Thus thequestionfailstohelpwherehelpismostneeded.

(2)Ifthesuggestionisunderstood,itgivesthewholesecretaway,verylittleremainsforthestudenttodo.

(3)The suggestion is of too special a nature. Even if the student canmake use of it in solving thepresentproblem,nothingislearnedforfutureproblems.Thequestionisnotinstructive.

(4)Evenifheunderstandsthesuggestion,thestudentcanscarcelyunderstandhowtheteachercametothe ideaofputting suchaquestion.Andhowcouldhe, the student, find suchaquestionbyhimself? Itappearsasanunnaturalsurprise,asarabbitpulledoutofahat;itisreallynotinstructive.

Noneoftheseobjectionscanberaisedagainsttheproceduredescribedinsection10,oragainstthatinsection15.

MOREEXAMPLES18.Aproblemofconstruction. Inscribea square inagiven triangle.Twoverticesof the square

shouldbeonthebaseofthetriangle,thetwootherverticesofthesquareonthetwoothersidesofthetriangle,oneoneach.“Whatistheunknown?”“Asquare.”“Whatarethedata?”“Atriangleisgiven,nothingelse.”“Whatisthecondition?”“Thefourcornersofthesquareshouldbeontheperimeterofthetriangle,twocornersonthebase,one

corneroneachoftheothertwosides.”“Isitpossibletosatisfythecondition?”“Ithinkso.Iamnotsosure.”“Youdonotseemtofindtheproblemtooeasy.Ifyoucannotsolvetheproposedproblem,trytosolve

firstsomerelatedproblem.Couldyousatisfyapartofthecondition?”“Whatdoyoumeanbyapartofthecondition?”“Yousee,theconditionisconcernedwithalltheverticesofthesquare.Howmanyverticesarethere?”“Four.”“Apart of the conditionwouldbe concernedwith less than four vertices.Keep only a part of the

condition,droptheotherpart.Whatpartoftheconditioniseasytosatisfy?”“Itiseasytodrawasquarewithtwoverticesontheperimeterofthetriangle—orevenonewiththree

verticesontheperimeter!”“Drawafigure!”ThestudentdrawsFig.2.“Youkeptonlyapartofthecondition,andyoudroppedtheotherpart.Howfaristheunknownnow

determined?”

FIG.2

“Thesquareisnotdeterminedifithasonlythreeverticesontheperimeterofthetriangle.”“Good!Drawafigure.”ThestudentdrawsFig.3.

FIG.3

“Thesquare,asyousaid,isnotdeterminedbythepartoftheconditionyoukept.Howcanitvary?”.....“Threecornersofyoursquareareontheperimeterofthetrianglebutthefourthcornerisnotyetthere

whereitshouldbe.Yoursquare,asyousaid,isundetermined,itcanvary;thesameistrueofitsfourthcorner.Howcanitvary?”

.....“Tryitexperimentally,ifyouwish.Drawmoresquareswiththreecornersontheperimeterinthesame

wayasthetwosquaresalreadyinthefigure.Drawsmallsquaresandlargesquares.Whatseemstobethelocusofthefourthcorner?Howcanitvary?”

Theteacherbroughtthestudentveryneartotheideaofthesolution.Ifthestudentisabletoguessthatthelocusofthefourthcornerisastraightline,hehasgotit.

19.Aproblemtoprove.Twoanglesareindifferentplanesbuteachsideofoneisparalleltothecorrespondingsideoftheother,andhasalsothesamedirection.Provethatsuchanglesareequal.

Whatwehavetoproveisafundamentaltheoremofsolidgeometry.Theproblemmaybeproposedtostudentswho are familiarwith plane geometry and acquaintedwith those few facts of solid geometrywhichpreparethepresenttheoreminEuclid’sElements.(Thetheoremthatwehavestatedandaregoingtoproveistheproposition10ofBookXIofEuclid.)Notonlyquestionsandsuggestionsquotedfromourlistareprintedinitalicsbutalsootherswhichcorrespondtothemas“problemstoprove”correspondto

“problemstofind.”(ThecorrespondenceisworkedoutsystematicallyinPROBLEMSTOFIND,PROBLEMSTOPROVE5,6.)“Whatisthehypothesis?”“Twoanglesareindifferentplanes.Eachsideofoneisparalleltothecorrespondingsideoftheother,

andhasalsothesamedirection.“Whatistheconclusion?”“Theanglesareequal.”“Drawafigure.Introducesuitablenotation.”ThestudentdrawsthelinesofFig.4andchooses,helpedmoreorlessbytheteacher,thelettersasin

Fig.4.“Whatisthehypothesis?Sayit,please,usingyournotation.”“A,B,C are not in the same plane asA′, B′, C′. AndAB ||A′B′, AC ||A′C′. AlsoAB has the same

directionasA′B′,andACthesameasA′C′.”

FIG.4

“Whatistheconclusion?”“∠BAC=∠B′A′C′.”“Look at the conclusion! And try to think of a familiar theorem having the same or a similar

conclusion.”“Iftwotrianglesarecongruent,thecorrespondinganglesareequal.”“Verygood!Nowhereisatheoremrelatedtoyoursandprovedbefore.Couldyouuseit?”“IthinksobutIdonotseeyetquitehow.”“Shouldyouintroducesomeauxiliaryelementinordertomakeitsusepossible?”.....“Well, the theoremwhichyouquotedsowell isabout triangles,aboutapairofcongruent triangles.

Haveyouanytrianglesinyourfigure?”“No.ButIcouldintroducesome.LetmejoinBtoC,andB′toC′.Thentherearetwotriangles,ΔABC,

ΔA′B′C′.”“Welldone.Butwhatarethesetrianglesgoodfor?”

“Toprovetheconclusion,∠BAC=∠B′A′C′.”“Good!Ifyouwishtoprovethis,whatkindoftrianglesdoyouneed?”

FIG.5

“Congruenttriangles.Yes,ofcourse,ImaychooseB,C,B′,C′sothat

AB=A′B′,AC=A′C′.”

“Verygood!Now,whatdoyouwishtoprove?”“Iwishtoprovethatthetrianglesarecongruent,

ΔABC=ΔA′B′C′.

IfIcouldprovethis,theconclusion∠BAC=∠B′A′C′wouldfollowimmediately.”“Fine!Youhaveanewaim,youaimatanewconclusion.Lookattheconclusion!Andtrytothinkof

afamiliartheoremhavingthesameorasimilarconclusion.”“Twotrianglesarecongruentif—ifthethreesidesoftheoneareequalrespectivelytothethreesides

oftheother.”“Welldone.Youcouldhavechosenaworseone.Nowhereisatheoremrelatedtoyoursandproved

before.Couldyouuseit?”“IcoulduseitifIknewthatBC=B′C′.”“Thatisright!Thus,whatisyouraim?”“ToprovethatBC=B′C′.”“Trytothinkofafamiliartheoremhavingthesameorasimilarconclusion.”“Yes,Iknowatheoremfinishing:‘...thenthetwolinesareequal.’Butitdoesnotfitin.”“Shouldyouintroducesomeauxiliaryelementinordertomakeitsusepossible?”.....“Yousee,howcouldyouproveBC=B′C′whenthereisnoconnectioninthefigurebetweenBCandB

′C′?”“Didyouusethehypothesis?Whatisthehypothesis?”

“WesupposethatAB||A′B′,AC||A′C′.Yes,ofcourse,Imustusethat.”“Did you use thewhole hypothesis? You say thatAB ||A′B′. Is that all that you know about these

lines?”“No;ABisalsoequaltoA′B′,byconstruction.Theyareparallelandequaltoeachother.Andsoare

ACandA′C′.”“Twoparallellinesofequallength—itisaninterestingconfiguration.Haveyouseenitbefore?”“Ofcourse!Yes!Parallelogram!LetmejoinAtoA′,BtoB′,andCtoC′.”“Theideaisnotsobad.Howmanyparallelogramshaveyounowinyourfigure?”“Two.No, three.No, two. Imean, thereare twoofwhichyoucanprove immediately that they are

parallelograms.Thereisathirdwhichseemstobeaparallelogram;IhopeIcanprovethatitisone.Andthentheproofwillbefinished!”

Wecouldhavegathered fromhis foregoinganswers that the student is intelligent.Butafter this lastremarkofhis,thereisnodoubt.

Thisstudentisabletoguessamathematicalresultandtodistinguishclearlybetweenproofandguess.He knows also that guesses can be more or less plausible. Really, he did profit something from hismathematics classes; he has some real experience in solving problems, he can conceive and exploit agoodidea.

20.Arateproblem.Waterisflowingintoaconicalvesselattherater.Thevesselhastheshapeofarightcircularcone,withhorizontalbase,thevertexpointingdownwards;theradiusofthebaseisa,thealtitudeoftheconeb.Findtherateatwhichthesurfaceisrisingwhenthedepthofthewaterisy.Finally,obtainthenumericalvalueoftheunknownsupposingthata=4ft.,b=3ft.,r=2cu.ft.perminute,andy=1ft.

FIG.6

The students are supposed to know the simplest rules of differentiation and the notion of “rate ofchange.”“Whatarethedata?”“Theradiusof thebaseof theconea=4ft., thealtitudeof theconeb=3 ft., the rateatwhich the

waterisflowingintothevesselr=2cu.ft.perminute,andthedepthofthewateratacertainmoment,y=1ft.”

“Correct.Thestatementoftheproblemseemstosuggestthatyoushoulddisregard,provisionally,thenumericalvalues,workwiththeletters,expresstheunknownintermsofa,b,r,yandonlyfinally,afterhavingobtainedtheexpressionoftheunknowninletters,substitutethenumericalvalues.Iwouldfollow

thissuggestion.Now,whatistheunknown?”“Therateatwhichthesurfaceisrisingwhenthedepthofthewaterisy.”“Whatisthat?Couldyousayitinotherterms?”“Therateatwhichthedepthofthewaterisincreasing.”“Whatisthat?Couldyourestateitstilldifferently?”“Therateofchangeofthedepthofthewater.”“Thatisright,therateofchangeofy.Butwhatistherateofchange?Gobacktothedefinition.”“Thederivativeistherateofchangeofafunction.”“Correct.Now,isya function?Aswesaidbefore,wedisregard thenumericalvalueofy.Canyou

imaginethatychanges?”“Yes,y,thedepthofthewater,increasesasthetimegoesby.”“Thus,yisafunctionofwhat?”“Ofthetimet.”“Good.Introducesuitablenotation.Howwouldyouwritethe‘rateofchangeofy’ inmathematical

symbols?”

“Good.Thus, this isyourunknown.Youhavetoexpressit intermsofa,b,r,y.By theway,oneofthesedataisa‘rate.’Whichone?”“ristherateatwhichwaterisflowingintothevessel.”“Whatisthat?Couldyousayitinotherterms?”“ristherateofchangeofthevolumeofthewaterinthevessel.”“Whatisthat?Couldyourestateitstilldifferently?Howwouldyouwriteitinsuitablenotation?”

“WhatisV?”“Thevolumeofthewaterinthevesselatthetimet.”

“Good.Thus,youhavetoexpress intermsofa,b, ,y.Howwillyoudoit?”

.....“Ifyoucannotsolvetheproposedproblemtrytosolvefirstsomerelatedproblem.Ifyoudonotsee

yettheconnectionbetween andthedata,trytobringinsomesimplerconnectionthatcouldserveasa

steppingstone.”.....“Doyounotseethatthereareotherconnections?Forinstance,areyandVindependentofeachother?”“No.Whenyincreases,Vmustincreasetoo.”“Thus,thereisaconnection.Whatistheconnection?”“Well,V is thevolumeofaconeofwhich thealtitude isy.But Idonotknowyet the radiusof the

base.”“Youmayconsiderit,nevertheless.Callitsomething,sayx.”

“Correct.Now,whataboutx?Isitindependentofy?”“No.Whenthedepthofthewater,y,increasestheradiusofthefreesurface,x,increasestoo.”“Thus,thereisaconnection.Whatistheconnection?”“Ofcourse,similartriangles.

x:y=a:b.”

“Onemoreconnection,yousee.Iwouldnotmissprofitingfromit.Donotforget,youwishedtoknowtheconnectionbetweenVandy.”

“Ihave

“Verygood.Thislookslikeasteppingstone,doesitnot?Butyoushouldnotforgetyourgoal.Whatistheunknown?”

“Well, .”

“You have to find a connection between , , and other quantities. And here you have one

betweeny,V,andotherquantities.Whattodo?”“Differentiate!Ofcourse!

Hereitis.”“Fine!Andwhataboutthenumericalvalues?”

“Ifa=4,b=3, =r=2,y=1,then

PARTII.HOWTOSOLVEITADIALOGUE

GettingAcquainted

WhereshouldIstart?Startfromthestatementoftheproblem.WhatcanIdo?Visualizetheproblemasawholeasclearlyandasvividlyasyoucan.Donotconcern

yourselfwithdetailsforthemoment.WhatcanIgainbydoingso?Youshouldunderstandtheproblem,familiarizeyourselfwithit,impress

itspurposeonyourmind.Theattentionbestowedon theproblemmayalsostimulateyourmemoryandpreparefortherecollectionofrelevantpoints.

WorkingforBetterUnderstandingWhereshouldIstart?Startagainfromthestatementof theproblem.Startwhen thisstatement isso

cleartoyouandsowellimpressedonyourmindthatyoumaylosesightofitforawhilewithoutfearoflosingitaltogether.WhatcanIdo?Isolatetheprincipalpartsofyourproblem.Thehypothesisandtheconclusionarethe

principalpartsofa“problemtoprove”;theunknown,thedata,andtheconditionsaretheprincipalpartsof a “problem to find.” Go through the principal parts of your problem, consider them one by one,consider them in turn, consider them in various combinations, relating each detail to other details andeachtothewholeoftheproblem.WhatcanIgainbydoingso?Youshouldprepareandclarifydetailswhicharelikelytoplayarole

afterwards.

HuntingfortheHelpfulIdeaWhereshouldIstart?Startfromtheconsiderationoftheprincipalpartsofyourproblem.Startwhen

these principal parts are distinctly arranged and clearly conceived, thanks to your previouswork, andwhenyourmemoryseemsresponsive.What can I do? Consider your problem from various sides and seek contacts with your formerly

acquiredknowledge.Consider your problem from various sides. Emphasize different parts, examine different details,

examinethesamedetailsrepeatedlybutindifferentways,combinethedetailsdifferently,approachthemfromdifferentsides.Trytoseesomenewmeaningineachdetail,somenewinterpretationofthewhole.

Seek contacts with your formerly acquired knowledge. Try to think of what helped you in similarsituationsinthepast.Trytorecognizesomethingfamiliarinwhatyouexamine,trytoperceivesomethingusefulinwhatyourecognize.WhatcouldIperceive?Ahelpfulidea,perhapsadecisiveideathatshowsyouataglancethewayto

theveryend.Howcananideabehelpful?Itshowsyouthewholeofthewayorapartoftheway;itsuggeststoyou

moreorlessdistinctlyhowyoucanproceed.Ideasaremoreorlesscomplete.Youareluckyifyouhaveanyideaatall.WhatcanIdowithanincompleteidea?Youshouldconsiderit.Ifitlooksadvantageousyoushould

consider it longer. If it looks reliable you should ascertain how far it leads you, and reconsider thesituation.Thesituationhaschanged,thankstoyourhelpfulidea.Considerthenewsituationfromvarioussidesandseekcontactswithyourformerlyacquiredknowledge.WhatcanIgainbydoingsoagain?Youmaybeluckyandhaveanotheridea.Perhapsyournextidea

willleadyoutothesolutionrightaway.Perhapsyouneedafewmorehelpfulideasafterthenext.Perhapsyouwillbeledastraybysomeofyourideas.Neverthelessyoushouldbegratefulforallnewideas,alsoforthelesserones,alsoforthehazyones,alsoforthesupplementaryideasaddingsomeprecisiontoahazyone,orattempting thecorrectionofa less fortunateone.Even ifyoudonothaveanyappreciablenewideasforawhileyoushouldbegratefulifyourconceptionoftheproblembecomesmorecompleteormorecoherent,morehomogeneousorbetterbalanced.

CarryingOutthePlanWhereshouldIstart?Startfromtheluckyideathatledyoutothesolution.Startwhenyoufeelsureof

yourgraspofthemainconnectionandyoufeelconfidentthatyoucansupplytheminordetailsthatmaybewanting.WhatcanIdo?Makeyourgraspquitesecure.Carrythroughindetailallthealgebraicorgeometric

operationswhich you have recognized previously as feasible.Convince yourself of the correctness ofeachstepbyformalreasoning,orby intuitive insight,orbothways ifyoucan. Ifyourproblemisverycomplexyoumaydistinguish“great”stepsand“small”steps,eachgreatstepbeingcomposedofseveralsmallones.Checkfirstthegreatsteps,andgetdowntothesmalleronesafterwards.What can I gainbydoing so?Apresentationof the solution each stepofwhich is correct beyond

doubt.

LookingBackWhereshouldIstart?Fromthesolution,completeandcorrectineachdetail.What can I do? Consider the solution from various sides and seek contacts with your formerly

acquiredknowledge.Considerthedetailsofthesolutionandtrytomakethemassimpleasyoucan;surveymoreextensive

partsofthesolutionandtrytomakethemshorter;trytoseethewholesolutionataglance.Trytomodifyto theiradvantagesmalleror largerpartsof thesolution, try to improve thewholesolution, tomake itintuitive, tofit it intoyourformerlyacquiredknowledgeasnaturallyaspossible.Scrutinizethemethodthatledyoutothesolution,trytoseeitspoint,andtrytomakeuseofitforotherproblems.Scrutinizetheresultandtrytomakeuseofitforotherproblems.Whatcan Igainbydoingso?Youmay findanewandbetter solution,youmaydiscovernewand

interestingfacts.Inanycase,ifyougetintothehabitofsurveyingandscrutinizingyoursolutionsinthisway,youwillacquiresomeknowledgewellorderedandreadytouse,andyouwilldevelopyourabilityofsolvingproblems.

PARTIII.SHORTDICTIONARYOFHEURISTIC

Analogy is a sort of similarity. Similar objects agree with each other in some respect, analogousobjectsagreeincertainrelationsoftheirrespectiveparts.

1. A rectangular parallelogram is analogous to a rectangular parallelepiped. In fact, the relationsbetweenthesidesoftheparallelogramaresimilartothosebetweenthefacesoftheparallelepiped:

Eachsideoftheparallelogramisparalleltojustoneotherside,andisperpendiculartotheremainingsides.

Eachfaceoftheparallelepipedisparalleltojustoneotherface,andisperpendiculartotheremainingfaces.

Letusagreetocallasidea“boundingelement”oftheparallelogramandafacea“boundingelement”oftheparallelepiped.Then,wemaycontractthetwoforegoingstatementsintoonethatappliesequallytobothfigures:

Each bounding element is parallel to just one other bounding element and is perpendicular to theremainingboundingelements.

Thus, we have expressed certain relations which are common to the two systems of objects wecompared,sidesoftherectangleandfacesoftherectangularparallelepiped.Theanalogyofthesesystemsconsistsinthiscommunityofrelations.

2. Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well asartisticways of expression and the highest scientific achievements.Analogy is used on very differentlevels.Peopleoftenusevague,ambiguous,incomplete,orincompletelyclarifiedanalogies,butanalogymayreachthelevelofmathematicalprecision.Allsortsofanalogymayplayaroleinthediscoveryofthesolutionandsoweshouldnotneglectanysort.

3.Wemay consider ourselves luckywhen, trying to solve a problem,we succeed in discovering asimpleranalogousproblem. Insection15,ouroriginalproblemwasconcernedwith thediagonalofarectangular parallelepiped; the consideration of a simpler analogous problem, concerned with thediagonalofarectangle,ledustothesolutionoftheoriginalproblem.Wearegoingtodiscussonemorecaseofthesamesort.Wehavetosolvethefollowingproblem:Findthecenterofgravityofahomogeneoustetrahedron.Withoutknowledgeoftheintegralcalculus,andwithlittleknowledgeofphysics,thisproblemisnot

easyatall;itwasaseriousscientificprobleminthedaysofArchimedesorGalileo.Thus,ifwewishtosolveitwithaslittlepreliminaryknowledgeaspossible,weshouldlookaroundforasimpleranalogousproblem.Thecorrespondingproblemintheplaneoccursherenaturally:Findthecenterofgravityofahomogeneoustriangle.Now,wehavetwoquestionsinsteadofone.Buttwoquestionsmaybeeasiertoanswerthanjustone

question—providedthatthetwoquestionsareintelligentlyconnected.4.Layingaside,forthemoment,ouroriginalproblemconcerningthetetrahedron,weconcentrateupon

the simpler analogous problem concerning the triangle. To solve this problem, we have to knowsomethingaboutcentersofgravity.Thefollowingprincipleisplausibleandpresentsitselfnaturally.

IfasystemofmassesSconsistsofparts,eachofwhichhasitscenterofgravityinthesameplane,thenthisplanecontainsalsothecenterofgravityofthewholesystemS.

This principle yields all thatweneed in the case of the triangle.First, it implies that the center ofgravityofthetriangleliesintheplaneofthetriangle.Then,wemayconsiderthetriangleasconsistingoffibers(thinstrips,“infinitelynarrow”parallelograms)paralleltoacertainsideofthetriangle(thesideABinFig.7).Thecenterofgravityofeachfiber(ofanyparallelogram)is,obviously,itsmidpoint,andallthesemidpointslieonthelinejoiningthevertexCoppositetothesideABtothemidpointMofAB(seeFig.7).

FIG.7

AnyplanepassingthroughthemedianCMofthetrianglecontainsthecentersofgravityofallparallelfiberswhichconstitute the triangle.Thus,weare led to theconclusion that thecenterofgravityof thewholetriangleliesonthesamemedian.Yetitmustlieontheothertwomediansjustaswell,itmustbethecommonpointofintersectionofallthreemedians.

It isdesirabletoverifynowbypuregeometry, independentlyofanymechanicalassumption,that thethreemediansmeetinthesamepoint.

5. After the case of the triangle, the case of the tetrahedron is fairly easy.We have now solved aproblemanalogoustoourproposedproblemand,havingsolvedit,wehaveamodeltofollow.

Insolving theanalogousproblemwhichweusenowasamodel,weconceived the triangleABCasconsistingoffibersparalleltooneofitssides,AB.Now,weconceivethetetrahedronABCDasconsistingoffibersparalleltooneofitsedges,AB.

Themidpointsofthefiberswhichconstitutethetrianglelieallonthesamestraightline,amedianofthetriangle,joiningthemidpointMofthesideABtotheoppositevertexC.Themidpointsofthefiberswhichconstitute the tetrahedronlieall in thesameplane, joiningthemidpointMof theedgeAB to theoppositeedgeCD(seeFig.8);wemaycallthisplaneMCDamedianplaneofthetetrahedron.

FIG.8

Inthecaseofthetriangle,wehadthreemedianslikeMC,eachofwhichhastocontainthecenterofgravityofthetriangle.Therefore,thesethreemediansmustmeetinonepointwhichispreciselythecenterofgravity.Inthecaseof the tetrahedronwehavesixmedianplanes likeMCD, joining themidpointofsome edge to the opposite edge, each ofwhich has to contain the center of gravity of the tetrahedron.Therefore,thesesixmedianplanesmustmeetinonepointwhichispreciselythecenterofgravity.

6. Thus, we have solved the problem of the center of gravity of the homogeneous tetrahedron. Tocomplete our solution, it is desirable to verify now by pure geometry, independently of mechanicalconsiderations,thatthesixmedianplanesmentionedpassthroughthesamepoint.

Whenwehad solved theproblemof the centerofgravityof thehomogeneous triangle,we found itdesirabletoverify,inordertocompleteoursolution,thatthethreemediansofthetrianglepassthroughthesamepoint.Thisproblemisanalogoustotheforegoingbutvisiblysimpler.

Againwemayuse,insolvingtheproblemconcerningthetetrahedron,thesimpleranalogousproblemconcerningthetriangle(whichwemaysupposehereassolved).Infact,considerthethreemedianplanes,passing through the three edgesDA, DB, DC issued from the vertexD; each passes also through themidpointoftheoppositeedge(themedianplanethroughDCpassesthroughM,seeFig.8).Now, thesethree median planes intersect the plane of Δ ABC in the three medians of this triangle. These threemedianspassthroughthesamepoint(thisistheresultofthesimpleranalogousproblem)andthispoint,justasD,isacommonpointofthethreemedianplanes.Thestraightline,joiningthetwocommonpoints,iscommontoallthreemedianplanes.

Weprovedthatthose3amongthe6medianplaneswhichpassthroughthevertexDhaveacommonstraight line.The samemustbe trueof those3medianplaneswhichpass throughA; and alsoof the3medianplanesthroughB;andalsoofthe3throughC.Connectingthesefactssuitably,wemayprovethatthe 6median planes have a commonpoint. (The 3median planes passing through the sides ofΔABCdetermineacommonpoint,and3linesofintersectionwhichmeetinthecommonpoint.Now,bywhatwehavejustproved,througheachlineofintersectiononemoremedianplanemustpass.)

7.Bothunder5andunder6weusedasimpleranalogousproblem,concerningthetriangle,tosolveaproblemaboutthetetrahedron.Yetthetwocasesaredifferentinanimportantrespect.Under5,weusedthemethod of the simpler analogousproblemwhose solutionwe imitatedpointbypoint.Under6,weusedtheresultofthesimpleranalogousproblem,andwedidnotcarehowthisresulthadbeenobtained.

Sometimes,wemaybeable touseboth themethodand the result of the simpler analogousproblem.Evenourforegoingexampleshowsthisifweregardtheconsiderationsunder5and6asdifferentpartsofthesolutionofthesameproblem.

Our example is typical. In solving a proposed problem,we can often use the solution of a simpleranalogousproblem;wemaybeabletouseitsmethod,oritsresult,orboth.Ofcourse,inmoredifficultcases,complicationsmayarisewhicharenotyetshownbyourexample.Especially,itcanhappenthatthesolutionoftheanalogousproblemcannotbeimmediatelyusedforouroriginalproblem.Then,itmaybeworthwhiletoreconsiderthesolution,tovaryandtomodifyittill,afterhavingtriedvariousformsofthesolution,wefindeventuallyonethatcanbeextendedtoouroriginalproblem.

8. It isdesirable to foresee the result, or, at least, some featuresof the result,with somedegreeofplausibility.Suchplausibleforecastsareoftenbasedonanalogy.

Thus,wemayknowthatthecenterofgravityofahomogeneoustrianglecoincideswiththecenterofgravityofitsthreevertices(thatis,ofthreematerialpointswithequalmasses,placedintheverticesofthe triangle).Knowing this,wemayconjecture that thecenterofgravityofahomogeneous tetrahedroncoincideswiththecenterofgravityofitsfourvertices.

Thisconjectureisan“inferencebyanalogy.”Knowingthatthetriangleandthetetrahedronarealikeinmanyrespects,weconjecture that theyarealike inonemorerespect. Itwouldbefoolish to regard theplausibility of such conjectures as certainty, but it would be just as foolish, or evenmore foolish, todisregardsuchplausibleconjectures.

Inferencebyanalogyappears tobe themostcommonkindofconclusion,andit ispossibly themostessential kind. It yields more or less plausible conjectures which may or may not be confirmed byexperienceandstricterreasoning.Thechemist,experimentingonanimalsinordertoforeseetheinfluenceofhisdrugsonhumans,drawsconclusionsbyanalogy.ButsodidasmallboyIknew.Hispetdoghadtobetakentotheveterinary,andheinquired:

“Whoistheveterinary?”“Theanimaldoctor.”“Whichanimalistheanimaldoctor?”9. An analogical conclusion from many parallel cases is stronger than one from fewer cases. Yet

quality is still more important here than quantity. Clear-cut analogies weighmore heavily than vaguesimilarities,systematicallyarrangedinstancescountformorethanrandomcollectionsofcases.

Intheforegoing(under8)weputforwardaconjectureaboutthecenterofgravityofthetetrahedron.Thisconjecturewassupportedbyanalogy;thecaseofthetetrahedronisanalogoustothatofthetriangle.Wemaystrengthentheconjecturebyexaminingonemoreanalogouscase,thecaseofahomogeneousrod(thatis,astraightline-segmentofuniformdensity).

Theanalogybetween

segmenttriangletetrahedron

hasmanyaspects.Asegmentiscontainedinastraightline,atriangleinaplane,atetrahedroninspace.Straightline-segmentsarethesimplestone-dimensionalboundedfigures,trianglesthesimplestpolygons,tetrahedronsthesimplestpolyhedrons.

The segment has 2 zero-dimensional bounding elements (2 end-points) and its interior is one-dimensional.

The trianglehas3 zero-dimensional and3one-dimensionalboundingelements (3vertices, 3 sides)anditsinterioristwo-dimensional.

Thetetrahedronhas4zero-dimensional,6one-dimensional,and4two-dimensionalboundingelements(4vertices,6edges,4faces),anditsinterioristhree-dimensional.

These numbers can be assembled into a table.The successive columns contain the numbers for thezero-, one-, two-, and three-dimensional elements, the successive rows the numbers for the segment,triangle,andtetrahedron:

VerylittlefamiliaritywiththepowersofabinomialisneededtorecognizeinthesenumbersasectionofPascal’striangle.Wefoundaremarkableregularityinsegment,triangle,andtetrahedron.

10. If we have experienced that the objects we compare are closely connected, “inferences byanalogy,”asthefollowing,mayhaveacertainweightwithus.

Thecenterofgravityofahomogeneousrodcoincideswith thecenterofgravityof its2end-points.The center of gravity of a homogeneous triangle coincideswith the center of gravity of its 3 vertices.Shouldwenotsuspectthatthecenterofgravityofahomogeneoustetrahedroncoincideswiththecenterofgravityofits4vertices?

Again,thecenterofgravityofahomogeneousroddividesthedistancebetweenitsend-pointsintheproportion 1 : 1. The center of gravity of a triangle divides the distance between any vertex and themidpointoftheoppositesideintheproportion2:1.Shouldwenotsuspectthatthecenterofgravityofahomogeneoustetrahedrondividesthedistancebetweenanyvertexandthecenterofgravityoftheoppositefaceintheproportion3:1?

Itappearsextremelyunlikelythat theconjecturessuggestedbythesequestionsshouldbewrong, thatsuch a beautiful regularity should be spoiled. The feeling that harmonious simple order cannot bedeceitfulguidesthediscovererbothinthemathematicalandintheothersciences,andisexpressedbytheLatinsaying:simplexsigillumveri(simplicityisthesealoftruth).

[Theprecedingsuggestsanextensiontondimensions.Itappearsunlikelythatwhatistrueinthefirstthreedimensions, forn=1,2,3, shouldcease tobe true forhighervaluesofn.Thisconjecture is an“inferenceby induction”; it illustrates that induction isnaturallybasedonanalogy.See INDUCTION ANDMATHEMATICALINDUCTION.]

[11.We finish thepresent sectionbyconsideringbriefly themost important cases inwhichanalogyattainstheprecisionofmathematicalideas.

(I)Twosystemsofmathematicalobjects,saySandS′,aresoconnectedthatcertainrelationsbetweentheobjectsofSaregovernedbythesamelawsasthosebetweentheobjectsofS′.

ThiskindofanalogybetweenSandS′isexemplifiedbywhatwehavediscussedunder1;takeasSthesidesofarectangle,asS′thefacesofarectangularparallelepiped.

(II)There isaone-onecorrespondencebetween theobjectsof the twosystemsSandS′,preservingcertain relations.That is, if sucha relationholdsbetween theobjectsofonesystem, thesame relationholdsbetweenthecorrespondingobjectsoftheothersystem.Suchaconnectionbetweentwosystemsisaveryprecisesortofanalogy;itiscalledisomorphism(orholohedralisomorphism).

(III)Thereisaone-manycorrespondencebetweentheobjectsofthetwosystemsSandS′preservingcertain relations.Suchaconnection (which is important invariousbranchesofadvancedmathematicalstudy,especiallyintheTheoryofGroups,andneednotbediscussedhereindetail)iscalledmerohedralisomorphism (or homomorphism; homoiomorphism would be, perhaps, a better term). Merohedralisomorphismmaybeconsideredasanotherveryprecisesortofanalogy.]

Auxiliaryelements.Thereismuchmoreinourconceptionoftheproblemattheendofourworkthan

wasinitaswestartedworking(PROGRESSANDACHIEVEMENT,1).Asourworkprogresses,weaddnewelementstothoseoriginallyconsidered.Anelementthatweintroduceinthehopethatitwillfurtherthesolutioniscalledanauxiliaryelement.

1.Therearevariouskindsofauxiliaryelements.Solvingageometricproblem,wemayintroducenewlines into our figure, auxiliary lines. Solving an algebraic problem, we may introduce an auxiliaryunknown(AUXILIARYPROBLEMS,1).Anauxiliarytheoremisatheoremwhoseproofweundertakeinthehopeofpromotingthesolutionofouroriginalproblem.

2.Therearevariousreasonsforintroducingauxiliaryelements.Wearegladwhenwehavesucceededin recollecting a problem related to ours and solved before. It is probable that we can use such aproblembutwedonotknowyethowtouseit.Forinstance,theproblemwhichwearetryingtosolveisageometricproblem,and the relatedproblemwhichwehavesolvedbeforeandhavenowsucceeded inrecollectingisaproblemabouttriangles.Yetthereisnotriangleinourfigure;inordertomakeanyuseoftheproblemrecollectedwemusthaveatriangle;therefore,wehavetointroduceone,byaddingsuitableauxiliarylinestoourfigure.Ingeneral,havingrecollectedaformerlysolvedrelatedproblemandwishingtouseitforourpresentone,wemustoftenask:Shouldweintroducesomeauxiliaryelementinordertomakeitsusepossible?(Theexampleinsection10istypical.)Goingbacktodefinitions,wehaveanotheropportunitytointroduceauxiliaryelements.Forinstance,

explicatingthedefinitionofacircleweshouldnotonlymentionitscenteranditsradius,butweshouldalsointroducethesegeometricelementsintoourfigure.Withoutintroducingthem,wecouldnotmakeanyconcreteuseofthedefinition;statingthedefinitionwithoutdrawingsomethingismerelip-service.

Tryingtouseknownresultsandgoingbacktodefinitionsareamongthebestreasonsforintroducingauxiliaryelements;buttheyarenottheonlyones.Wemayaddauxiliaryelementstotheconceptionofourproblem in order to make it fuller, more suggestive, more familiar although we scarcely know yetexplicitlyhowweshallbeabletousetheelementsadded.Wemayjustfeelthatitisa“brightidea”toconceivetheproblemthatwaywithsuchandsuchelementsadded.

Wemayhavethisorthatreasonforintroducinganauxiliaryelement,butweshouldhavesomereason.Weshouldnotintroduceauxiliaryelementswantonly.

3.Example.Constructatriangle,beinggivenoneangle,thealtitudedrawnfromthevertexofthegivenangle,andtheperimeterofthetriangle.

FIG.9

FIG.10

We introduce suitable notation. Letα denote the given angle, h the given altitude drawn from thevertexAofαandp thegivenperimeter.Wedrawafigure inwhichweeasilyplaceαandh.Haveweused all the data? No, our figure does not contain the given length p, equal to the perimeter of thetriangle.Thereforewemustintroducep.Buthow?

Wemayattempttointroducepinvariousways.TheattemptsexhibitedinFigs.9,10appearclumsy.Ifwetrytomakecleartoourselveswhytheyappearsounsatisfactory,wemayperceivethatitisforlackofsymmetry.

Infact, the trianglehas threeunknownsidesa,b,c.Wecalla,asusual, thesideopposite toA; weknowthat