Professor Higgins's Problem Collection PDF
Preview Professor Higgins's Problem Collection
PROFESSORHIGGINS’SPROBLEMCOLLECTION Professor Higgins’s Problem Collection PETER M. HIGGINS 3 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©PeterM.Higgins2017 Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2017 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2016957995 ISBN978–0–19–875547–0 Printedandboundby CPILitho(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. PREFACE Mathematicsisawonderfulsubjectbecauseitscontentissorich,diverse,anddeepbutwhat ismore,everythinginitistrue.Thatiswhystudentsenjoyapplyingwhattheyhavelearntto awidevarietyofquestions.Thefiftyproblemsherearepresentedunderfivebroadheadings: Number, Algebra, Geometry, Chance and combinations, and Movement. These wide fields covermostofthemathematicsmetatsecondaryschoollevel.Althoughtheemphasisison themathematicsitself,solvingtheproblemsallowsustocomprehendmanyinterestingfacets oftheworldfromencryption,tothemotionoftheplanets,tounderstandingwhyrecessive genetraitsneverdieout.Manyoftheprocessesweseegoingonaroundushaveunderlying mathematicalstructuresandsocannotbefullyunderstoodotherthanthroughmathematical analysis. Theproblemsaregenerallynotstandardexercisesbuthavesomedegreeofnoveltyorspe- cialinterestthatgoeswiththem.Theprimarypurposeistoallowyoutodosomethingwith yourmathematicsratherthanmerelypractiseit.Withineachpart,thereissomegradationin difficultyoftheproblems.Iwouldhopethatanyreadercouldlookatsomeoftheproblems andthinktothemselvesthattheycouldfigurethemoutsomehow. Thisbookdrawsfromawidescopeofmathematicalnotionsandforthatreasoneachpart beginswithashortsummaryofthekindsofquestionsthatyouareabouttoseewhilemore detailontherelevantmathematicalideasandnotationsinvolvedistobefoundattheclose ofeachpart.Theseremindersdonotconstituteacompletesummaryofthesubjectstouched upon,noraretheresultsstatedalwaysthemostgeneralpossible,buttheydocovertheideas thatcropupinthecourseofsolvingourproblems.Inthiswaythereadercanfind,withinthe bookitself,whatheorsheneedsforacompleteunderstandingofthesolutions. Asyouwillread,theformatconsistsofposingaquestion,orsometimesapairofrelated questions,thefullsolutiontowhichisrevealedbysimplyturningthepage.Atthecloseof eachpartthereisaconcludingsectionthatfillsinsomeofthebackgroundtotheproblems andoffersglimpsesastowheretheaccompanyingtrainsofthoughtlead.Thereismuchto discoverasyoureadthroughthebookandIhopethatyouwillenjoyitall. PeterM.Higgins UniversityofEssex ColchesterUK 2017 CONTENTS 1 Number 1 Introduction 1 Problem1:Whatvaluesmayprimestake? 3 Problem2:Hiddenbinary 5 Problem3:Howshouldwewritefractions? 7 Problem4:Labellingacube 9 Problem5:Mapnumberingproblem 11 Problem6:Countingthepennies 13 Problem7:Countingsolutions 15 Problem8:Crackingcodes 17 Problem9:Logicpuzzles 19 Problem10:Pigeonholetricks 21 Numbernotes 23 Commentsonnumberproblems 25 2 Algebra 27 Introduction 27 Problem11:Tradingequations 29 Problem12:Frogandtoad 31 Problem13:Geometricseriesquestions 33 Problem14:Summingpowersofintegers 35 Problem15:Rootsofequations 37 Problem16:Specialseries 39 Problem17:Exactvaluesoftrigonometricfunctions 41 Problem18:Cosinequestions 43 Problem19:Howmanytermstogetclose? 45 Problem20:Functionalequations 47 Algebranotes 50 Commentsonalgebraproblems 51 3 Geometry 53 Introduction 53 Problem21:Goatgrazingproblems 55 Problem22:Compassesandstraightedge 57 Problem23:Tinychangesmeanalot?Ornot? 59 Problem24:Goldendiagonal 61 Problem25:Newton’sladderproblem 63 viii CONTENTS Problem26:Howlongisafanbelt? 65 Problem27:Afifthcenturytrick 67 Problem28:Atatangent 69 Problem29:Lowlights 71 Problem30:Rugbyplacekickingproblem 73 Geometrynotes 76 Commentsongeometryproblems 78 4 Chanceandcombinations 79 Introduction 79 Problem31:Tennistournaments 81 Problem32:Tumblingdiceandatrickytoss 83 Problem33:Tactilerandomwalks 85 Problem34:Moreamblings 87 Problem35:Oneinamillion? 89 Problem36:LoyaltyversusSwitch 91 Problem37:Thinkit’sgoingtoraintoday? 93 Problem38:Persistenceofrecessivegenes 95 Problem39:Countingmappings 97 Problem40:Areaunderthebellcurve 99 Chanceandcombinationsnotes 101 Commentsonchanceandcombinationsproblems 101 5 Movement 103 Introduction 103 Problem41:Cyclingquestions 105 Problem42:Ontherun 107 Problem43:SamLoyd’sferryboatproblem 109 Problem44:Cylinderproblems 111 Problem45:CatchingTheGoblin 113 Problem46:Flaginthewind 115 Problem47:Planetarymotion 117 Problem48:Elasticcollision 119 Problem49:Slipslidingaway? 121 Problem50:Fermat’sPrincipleandSnell’sLawofRefraction 123 Movementnotes 125 Commentsonmovementproblems 126 PART 1 Number Introduction This problem set involves only ordinary counting numbers 1,2,3,··· and their related fractions.Webeginwithquestionsaboutprimenumbers,thenapairofproblemsaboutrep- resenting numbers as sums of particular types. Problems 4 to 7 concern counting up how manywaysthereareofdoingsomething,whileProblems8amd9areaboutlogicaldeduction. Problem8isaboutcodebreakingwhileProblem9involvescunninglygleaninginformationin ordertoeliminatepossibilities.Thefinalproblemisaboutexploitingapieceofmathematical commonsense,whichisthatifwetrytoputtoomanythingsintotoofewplacestherehasto besomedoublingup. Since prime numbers, factorization properties, and counting formulae arise in our solu- tions,thereisashortsummaryofthesetopicsattheendofthechapterthattellsyouwhatis neededintacklingtheseparticularquestions.
The list of books you might like
