Professor Higgins's Problem Collection

135 Pages·2017·2.936 MB·English

by  Peter M. Higgins

PROFESSORHIGGINS’SPROBLEMCOLLECTION

Professor Higgins’s Problem Collection
PETER M. HIGGINS
3
3
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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.