数学专业英语14.docx
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数学专业英语14.docx
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数学专业英语14
MathematicalEnglish
Dr.XiaominZhang
Email:
zhangxiaomin@
MathematicalPuzzles
Welcometomyselectionofmathematicalpuzzles.
Themathpuzzlespresentedhereareselectedforthedeceptivesimplicityoftheirstatement,ortheeleganceoftheirsolution. Theyrangeovergeometry,probability,numbertheory,algebra,calculus,trigonometry,andlogic. Allrequireacertainingenuity,butusuallyonlypre-collegemath. Somepuzzlesareoriginal.
Explaininghowananswerisarrivedatismoreimportantthantheansweritself. Tothisend,hints,answers,andfullyworkedsolutionsareprovided,togetherwithlinkstorelatedmathematicaltopics. Furtherreferencesareprovidedwithmanyofthesolutions. Thepuzzlesareintendedtobefun,withaneducationalelement.
1.Triangulararea
IntriangleABC,producealinefromBtoAC,meetingatD,andfromCtoAB,meetingatE. LetBDandCEmeetat X.
LetBXEhaveareaa,BXChaveareab,andCXDhaveareac. FindtheareaofquadrilateralAEXDintermsof a,b,and c.
Hint:
Considerpairsoftriangles,withcommonheightandcolinearbase.
Solution
BXEhasareaa,BXChasareab,andCXDhasareac.
Wewillusethefactthattheareaofatriangleisequalto½×base×perpendicularheight. Anysidecanserveasthebase,andthentheperpendicularheightextendsfromthevertexoppositethebasetomeetthebase(oranextensionofit)atrightangles.
ConsiderBXEandBXC,EX/XC=a/b.Similarly,consideringBXCandCXD,BX/XD=b/c.
NowdrawlineAX. LetAXEhaveareaxandAXDhaveareay.SinceAXBandAXDhavecommonheight,wehave(a+x)/y=b/c.
Similarly,x/(y+c)=a/b.
Cross-multiplyingyields:
by=cx+ac,bx=ay+ac.
Solvingthesesimultaneousequations,weobtainx=ac(a+b)/(b2-ac),y = ac(b + c)/(b2 - ac).
ThereforetheareaofquadrilateralAEXDis
.
2.Twologicians
Twoperfectlogicians,SandP,aretoldthatintegersxandyhavebeenchosensuchthat1 < x < yandx+y < 100. Sisgiventhevaluex+yandPisgiventhevaluexy. Theythenhavethefollowingconversation.
P:
Icannotdeterminethetwonumbers.
S:
Iknewthat.
P:
NowIcandeterminethem.
S:
SocanI.
Giventhattheabovestatementsaretrue,whatarethetwonumbers?
(Computerassistanceallowed.)
Hint:
Firstofall,trivially,xycannotbeprime. Italsocannotbethesquareofaprime,forthatwouldimplyx = y.
Wenowdeduceasmuchaspossiblefromeachofthelogicians'statements. Wehaveonlypublicinformation:
theproblemstatement,thelogicians'statements,andtheknowledgethatthelogicians,beingperfect,willalwaysmakecorrectandcompletedeductions. Eachlogicianhas,inaddition,onepieceofprivateinformation:
sumorproduct.
Solution
P:
Icannotdeterminethetwonumbers.
P'sstatementimpliesthatxycannothaveexactlytwodistinctproperfactorswhosesumislessthan 100. Callsuchapairoffactorseligible.
Forexample,xycannotbetheproductoftwodistinctprimes,forthenPcoulddeducethenumbers. Likewise,xycannotbethecubeofaprime,suchas33 = 27,forthen3×9wouldbeauniquefactorization;orthefourthpowerofaprime.
Othercombinationsareruledoutbythefactthatthesumofthetwofactorsmustbelessthan 100. Forexample,xycannotbe242 = 2×112,since11×22istheuniqueeligiblefactorization;2×121beingineligible. Similarlyforxy = 318 = 2×3×53.
S:
Iknewthat.
IfSwassurethatPcouldnotdeducethenumbers,thennoneofthepossiblesummandsofx+ycanbesuchthattheirproducthasexactlyonepairofeligiblefactors. Forexample,x+ycouldnotbe 51,sincesummands 17and 34producexy = 578,whichwouldpermitPtodeducethenumbers.
Wecangeneratealistofvaluesofx+ythatareneverthesumofpreciselytwoeligiblefactors. ThefollowinglistisgeneratedbyJavaScript;thefunctionmaybeinspectedbyviewingJavaScript:
functiongenSum(plaintext.)
Eligiblesums:
11,17,23,27,29,35,37,41,47,53.
(WecanuseGoldbach'sConjecture,whichstatesthateveryevenintegergreaterthan2canbeexpressedasthesumoftwoprimes,todeducethattheabovelistcancontainonlyoddnumbers. Althoughtheconjectureremainsunproven,ithasbeenconfirmedempiricallyupto4×1014.)
P:
NowIcandeterminethem.
Pnowknowsthatx+yisoneofthevalueslistedabove. IfthisenablesPtodeducexandy,then,oftheeligiblefactorizationsofxy,theremustbepreciselyoneforwhichthesumofthefactorsisinthelist. Thetablebelow,generatedbyJavaScript(viewplaintextJavaScript:
functiongenProd),showsallsuchxy,togetherwiththecorrespondingx,y,andx+y. Thetableissortedbysumandthenproduct.
Notethataproductmaybeabsentfromthetableforoneoftworeasons. Eithernoneofitseligiblefactorizationsappearsintheabovelistofeligiblesum
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