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ThePowerofGoodQuestionProvideby:BME1411/17Whatmakesagoodquestion?

Ifyou’redoingmathsforfun,orareaprofessionalmathematician,youanswerisgoingtobedifferent.（Aneasyquestionisboring）Ifyouareastudentfacingexams,youmight(understandably)saythatgoodmeanseasy.

2/17Wheredothesegoodquestionscomefrom?

Generalize

Simplifyandvary

Lookfornewtools

Takerisks3/17Generalizean

+

bn

=

cn?In

numbertheory,

Fermat'sLastTheorem

(sometimescalled

Fermat'sconjecture,especiallyinoldertexts)statesthatnothree

positiveintegers

a,

b,and

c

satisfytheequation

an

+

bn

=

cn

foranyintegervalueof

n

strictlygreaterthantwo.Thecases

n

=

1and

n

=

2havebeenknowntohaveinfinitelymanysolutionssinceantiquity.4/17350years!?

Thistheoremwasfirst

conjecturedby

PierredeFermat

in1637inthemarginofacopyof

Arithmetica(算術(shù))whereheclaimedhehadaproofthatwastoolargetofitinthemargin.

Thefirstsuccessfulproof

wasreleasedin1994by

AndrewWiles,andformallypublishedin1995,after358yearsofeffortbymathematicians..Itisamongthemostnotabletheoremsinthe

historyofmathematicsandpriortoitsproof,itwasinthe

GuinnessBookofWorldRecords

asthe"mostdifficultmathematicalproblem",oneofthereasonsbeingthatithasthelargestnumberofunsuccessfulproofs.5/17萬暢高清攝像機萬暢高清攝像機萬暢傳輸接入模塊萬暢局端模塊和設(shè)備視頻樞紐萬局端模塊Fermat’ssimplequestionturnedouttobeincrediblyfruitful:itgeneratednewmathematics,newinsightsandnewwaysoflookingatthings.Thoughhard,manymathematicianswouldregardthisasa“good”question.Togetherwith

RenéDescartes（笛卡爾）,Fermatwasoneofthetwoleadingmathematiciansofthefirsthalfofthe17thcentury.

6/17SimplifyandvaryGalleryproblemAniceexampleistheartgalleryproblem:howmanysecurityguardsdoyouneedtobesurethattogethertheycanoverseethewholeinteriorofanartgallery?7/17AnswerThefirstanswer,givenin1978fiveyearsafertheproblemwasposed.Usinganingeniouslineofattack,themathematicianS.Fiskprovedthatyouneverneedmorethan1/3guards,wherenisthenumberofvertices(corners)ofthepolygon.8/1730yearson,theseproblemisstillgoing

Whatiftheguardsarenotconfinedtothecornersofthegallery?

GalleryproblemsWhatiftheyareallowedtomovearound?

Whatifthereareobstaclesinthemiddleofthegallerythatyoucannotseethrough?

Thewallsarecurved?

Whatif,insteadofguardingatwo-dimensionalpolygon,youaretryingtoguardathree-dimensionalpolyhedron?

9/17LookfornewtoolsCalculusTherearealsoquestionsthatarebeingasked,notbyindividuals,butbyawholeage，cryingoutfornewmathematicaltools.Theiranswerscanspawnsomethingofarevolution.Agreatexampleistheinventionofcalculusintheseventeenthcentury.10/17CalculusHowcanwedescribecontinuouschange?Ajourney:speedistherateofchangeofdistancepertime,soyousimplydividethedistanceyoutraveledbythetimeittooktotravelit.(S/T)Butofcourse,youdidn‘ttravelatthataveragespeedateverymomentofyoujourney.Atsometimesyouwillhavebeengoingslowerandatsometimesfaster,withthespeedvaryingcontinuously.Toworkoutyourexactspeedataparticularmomentintime,youhavetocalculatetheinstantaneousrateofchangeofdistancewithrespecttotime.11/17ApplicationsofcalculusThemethodsfordoingthiswereinventedprimarilybyGottfriedLeibnizandIsaacApplicationsofintegralcalculusincludecomputationsinvolvingarea,volume,arclength,centerofmass,work,andpressure.MoreadvancedapplicationsincludepowerseriesandFourierseries.Calculusisalsousedtogainamorepreciseunderstandingofthenatureofspace,time,andmotion.GottfriedLeibniz(left)IsaacNewton(right)12/17TakerisksFourcolourtheoremNotallquestionsturnouttohaveinterestinganswers.Mathematicianssimplyhavetoaccepttheriskthataquestiontheychoosetoworkonmaynotbesolvedintheirlifetime,orthatitmayturnouttohaveaboringanswer.It‘sallpartofthecreativeprocess.Aquestionthatnotbesolvedintheirlifetime-Fermat’sLastTheorem.13/17FourcolortheoremItsaysthatfourcoloursareenoughtocolouramapdrawnontheplanesothatnotwoneighbouringcountrieshavethesamecolour.Theproofofthistheorem,whenitfinallycameinthe1970saftermathematicianshadbeenwrestlingwiththetheoremforoveracentury,wasdisappointing.Itusedabruteforceapproachinvolvingacomputercheckingthroughahugenumberofpossibilities,makingsuretheydidnotprovideacounterexampletothetheorem.Theapproachdeliverednonewinsightsatall.Asimplemapcolouredcorrectlywithfourcolours.14/17萬暢高清攝像機萬暢高清攝像機萬暢傳輸接入模塊萬暢局端模塊和設(shè)備視頻樞紐萬局端模塊However,thepartoftheirproofwasactuallydonebyacomputer.Nohumanbeingcouldintheirlifetimeeveractuallyreadtheentireprooftocheckthatitwascorrect.Severalmathematiciansofthetimecomplainedthatthismeantthatitwasn'treallyaproofatall!Ifnobodycouldchecktheproof,howcouldweeverknowwhetheritwasrightorwrong?Partoftheworldmap,colouredin