Table Of ContentP
r
THOMAS POVEY is Professor of In Professor Povey’s Perplexing Problems, Thomas
o
T H O M A S P O V E Y
Engineering Science at the University of ‘Entry to selective universities will often require students to demonstrate that f Povey shares 109 of his favourite problems
e
in physics and maths. A tour de force of
Oxford, where he researches jet-engine and they have engaged in super-curricular activities which develop their awareness s
s imagination and exposition, he takes us by the
rocket technology. His science puzzles have and understanding of their subject. Tom Povey has provided in this book o
hand and guides us through uncompromisingly
appeared in the Guardian newspaper and been enough super-curricular content to keep an aspiring mathematician, physicist, r
challenging territory that expands our minds and
featured on BBC Radio 4. engineer or material scientist (and their teachers) happy for months. For a P
o encourages a playful and exploratory approach
parent or teacher who wants to pro-actively support a student in preparing for
v to study. The puzzles, he says, are like toys. We
entry to a competitive science or maths course this is a no-brainer purchase.’ e
should pick up the one we most enjoy, and play
y
Mike Nicholson, Director of Undergraduate Admissions and
’s with it.
Outreach, University of Oxford (2006-14), Director of Student
P
Recruitment and Admissions, University of Bath e Whether you are an aspiring scientist or an old-
r
Professor hand, pitting yourself against these problems will
p
‘This book is about the excitement of mathematics and physics at the school/ l test your ability to think, and inspire you with
e
university boundary. It contains a remarkable collection of intriguing intellectual curiosity and enthusiasm for physics. Presented
x
challenges, all with complete solutions. You can read it to increase your ability i with charm and wit, the questions span the
n
in maths and physics; you can read it to develop your problem-solving skills; but g gap between high-school and university-
Povey ’s
entrance standard material. Detailed answers are
most of all you can read it for fun.’ Professor Richard Prager, Head of the P
School of Technology, University of Cambridge r lightened with a fascinating and refreshing blend
o of scientific history, application and personal
b
anecdote.
‘This instructive book will allow aspiring young l
e Perplexing
scientists to test and expand their knowledge of m
On this delightful and idiosyncratic romp
physics and mathematics, and gain greater s
through pre-university maths and physics, the
confidence in their abilities.’ Professor Sir Chris
T author shows us that behind every single one
Llewellyn Smith FRS, Director General H
of these questions lies a new way of thinking
Problems
of CERN (1994-98) O about subjects we thought we had understood.
M He argues that engaging with the unfamiliar is
A key to forming deeper insights and developing
S intellectual independence. Professor Povey’s
Perplexing Problems is a manifesto that science
P
O should be playful, and a celebration of the
V curious.
E
Y
ONEWORLD Pre-University Physics and Maths
MATHEMATICS / SCIENCE Puzzles with Solutions
wwwwww..oonneewwoorrlldd--ppuubblliiccaattiioonnss..ccoomm £25/$30
jacket illustrations courtesy of the author
PraiseforProfessorPovey’sPerplexingProblems
“Entrytoselectiveuniversitieswilloftenrequirestudentstodemonstratethat
theyhaveengagedinsuper-curricularactivitieswhichdeveloptheirawareness
andunderstandingoftheirsubject. ThomasPoveyhasprovidedinthisbook
enoughsuper-curricularcontenttokeepanaspiringmathematician,physicist,
engineer or material scientist (and their teachers) happy for months. For a
parentorteacherwhowantstoproactivelysupportastudentinpreparingfor
entrytoacompetitivescienceormathscoursethisisano-brainerpurchase.”
—MikeNicholson,DirectorofUndergraduateAdmissionsandOutreach,
University of Oxford (2006-14), Director of Student Recruitment and
Admissions,UniversityofBath
“Thisbookisabouttheexcitementofmathematicsandphysicsattheschool-
universityboundary. Itcontainsaremarkablecollectionofintriguingintellec-
tualchallenges, allwithcompletesolutions. Youcanreadittoincreaseyour
abilityinmathsandphysics;youcanreadittodevelopyourproblem-solving
skills;butmostofallyoucanreaditforfun.”
—Professor Richard Prager, Head of Cambridge University School of
Technology
“Thisinstructivebookwillallowaspiringyoungscientiststotestandexpand
their knowledge of physics and mathematics, and gain greater confidence in
theirabilities.”
—ProfessorSirChrisLlewellynSmithFRS,DirectorGeneralofCERN
(1994-98)
“Bychallengingstudentswithunfamiliarbutengagingproblems,IhopePro-
fessor Povey inspires many more budding physicists and mathematicians to
realisetheirpotential.”
—SirPeterLampl,FounderandChairmanoftheSuttonTrust
“Arealtourdeforceofscientificproblems. ThomasPoveyisclearlyamaster
atthiskindofconundrum... Fascinating.”
—JohnFarndon,authorofDoYouThinkYou’reClever?
“Thisbookwillamusethosewholikepuzzles. Thetopicsvaryfromtheprac-
ticaltothequirky. Readerswithabasicknowledgeofmathsandphysicswill
be entertained and encouraged by the interesting and challenging questions
theyareabletosolve.”
—ProfessorDameAnnDowlingDBEFRSFREng,PresidentoftheRoyal
AcademyofEngineering
Professor Povey’s
Perplexing Problems
Pre-University Physics and Maths Puzzles
with Solutions
Thomas Povey
AOneworldBook
FirstpublishedinNorthAmerica,GreatBritainandAustraliaby
OneworldPublications,2015
Copyright©ThomasPovey2015
ThemoralrightofThomasPoveytobeidentifiedastheAuthorofthisworkhasbeen
assertedbyhiminaccordancewiththeCopyright,Designs,andPatentsAct1988
Allrightsreserved
CopyrightundertheBerneConvention
ACIPrecordforthistitleisavailablefromtheBritishLibrary
ISBN978-1-78074-775-0
PrintedandboundbyCPIGroup(UK)Ltd,CroydonCR04YY
OneworldPublications
10BloomsburyStreet,LondonWC1B3SR,England
ToMikeLeaskandTerryJones—brilliantteachers,insatiableadventurers.
WiththankstoTetAmaya,forsubstantialcontributionsthroughout.
Contents
Introduction 1
Acknowledgements 5
Anoddjourney 7
1 Geometry 19
1.1 Shortestwalk(cid:70) . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2 Intercontinentaltelephonecable(cid:70) . . . . . . . . . . . . . . 21
1.3 Chessboardandhoop(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . 22
1.4 Hexagonaltilesandhoop(cid:70)(cid:70) . . . . . . . . . . . . . . . . . 24
1.5 Intersectingcircles(cid:70)(cid:70). . . . . . . . . . . . . . . . . . . . . 26
1.6 Cubewithinsphere(cid:70) . . . . . . . . . . . . . . . . . . . . . 28
1.7 Polygoninscribedwithincircle(cid:70) . . . . . . . . . . . . . . . 30
1.8 Circleinscribedwithinpolygon(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . 31
1.9 Triangleinscribedwithinsemicircle(cid:70) . . . . . . . . . . . . 34
1.10 Bigandsmalltreetrunks(cid:70)(cid:70)or(cid:70)(cid:70)(cid:70)(cid:70) . . . . . . . . . . 36
1.11 ProfessorFuddlethumbs’stamp(cid:70)(cid:70) . . . . . . . . . . . . . 39
1.12 CaptainFistfulls’treasure(cid:70) . . . . . . . . . . . . . . . . . . 41
1.13 CaptainFistfulls’treasureII(cid:70)(cid:70) . . . . . . . . . . . . . . . 42
1.14 CaptainFistfulls’treasureIII(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . 45
1.15 ThegeometryofKochIsland(cid:70)(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . 49
1.16 Aneasyishfencingproblem(cid:70) . . . . . . . . . . . . . . . . . 52
1.17 Ahardishfencingproblem(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . 54
2 Mathematics 60
2.1 Humancalculator(cid:70) . . . . . . . . . . . . . . . . . . . . . . 60
2.2 ProfessorFuddlethumbs’reports(cid:70) . . . . . . . . . . . . . . 63
2.3 MoreofProfessorFuddlethumbs’reports(cid:70) . . . . . . . . . 64
2.4 AntonacubeI(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . . 65
2.5 AntonacubeII(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . 66
2.6 AntonacubeIII(cid:70)(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . 67
2.7 Afallingraindrop(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . 69
2.8 TheThreeDoorProblem(cid:70)(cid:70). . . . . . . . . . . . . . . . . 72
vii
viii CONTENTS
2.9 DrBletchley’sPIN(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . 74
2.10 MrSmith’scoins(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . 76
2.11 Thethreeenvelopeproblem(cid:70). . . . . . . . . . . . . . . . . 77
2.12 Acardgame(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . . . . 79
3 Statics 82
3.1 Sewageworker’sconundrum(cid:70) . . . . . . . . . . . . . . . . 83
3.2 Sewageworker’sescape(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . 85
3.3 Sewageworker’sresolution(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . 89
3.4 Aztecstonemovers(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . 91
3.5 TheWheelWarsI(cid:70)(cid:70)(cid:70). . . . . . . . . . . . . . . . . . . . 96
3.6 TheWheelWarsII(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . 100
3.7 Obeliskraiser(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . . . 101
3.8 Obeliskrazer(cid:70)(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . 108
3.9 TheRavineof(NotQuite)CertainDeath(cid:70)(cid:70)(cid:70) . . . . . . 115
4 Dynamicsandcollisions 119
4.1 Pulleys(cid:70) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.2 DrLightspeed’selastotennismatch(cid:70)(cid:70) . . . . . . . . . . . . 124
4.3 Acceleratingmatchbox(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . 129
4.4 ThelastflightofMonsieurCanard(cid:70)(cid:70) . . . . . . . . . . . 132
4.5 Water-poweredfunicular(cid:70) . . . . . . . . . . . . . . . . . . 136
4.6 SherlockHolmesandtheBellaFioreemerald(cid:70)(cid:70)(cid:70) . . . . 141
4.7 Equivalentstatementsforlinearcollisions(cid:70)(cid:70)(cid:70) . . . . . . . 147
5 Circularmotion 150
5.1 Frictionatthesuperbikeraces(cid:70) . . . . . . . . . . . . . . . 151
5.2 Polepositionatthesuperbikeraces(cid:70)(cid:70). . . . . . . . . . . . 153
5.3 Rollercoaster(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . . 155
5.4 Derailedrollercoaster(cid:70)(cid:70). . . . . . . . . . . . . . . . . . . 158
5.5 ThelastrideofProfessorLazy(cid:70)(cid:70) . . . . . . . . . . . . . . 162
5.6 WallofDeath: car(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . 165
5.7 WallofDeath: motorcycle(cid:70)(cid:70)or(cid:70)(cid:70)(cid:70)(cid:70) . . . . . . . . . 170
6 Simpleharmonicmotion 176
6.1 Oscillatingsphere(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . 180
6.2 ProfessorStopclock’stime-manipulator(cid:70)(cid:70)(cid:70) . . . . . . . . 184
6.3 DrSpringlove’sOscillator(cid:70) . . . . . . . . . . . . . . . . . . 187
6.4 DrSpringlove’sInfernalOscillator(cid:70)(cid:70) . . . . . . . . . . . . 190
6.5 DrSpringlove’sImprovedInfernalOscillator(cid:70)(cid:70)(cid:70) . . . . . 192
7 Madinventionsandperpetualmotion 196
7.1 Stevin’sclootcrans(cid:70) . . . . . . . . . . . . . . . . . . . . . . 197
7.2 Power-producingspeedhumps(cid:70)(cid:70) . . . . . . . . . . . . . . 201
7.3 Theoverbalancedwheel(cid:70)(cid:70) . . . . . . . . . . . . . . . . . 203
7.4 ProfessorSinclair’ssyphon(cid:70)(cid:70) . . . . . . . . . . . . . . . . 209
CONTENTS ix
7.5 Boyle’sperpetualvase(cid:70) . . . . . . . . . . . . . . . . . . . . 214
7.6 Thecuriouswheel(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . 217
8 Kinematics 219
8.1 ProfessorLazy(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . . . 219
8.2 TheUnflinchingAviator(cid:70) . . . . . . . . . . . . . . . . . . 222
8.3 Targetshooting(cid:70) . . . . . . . . . . . . . . . . . . . . . . . 225
9 Electricity 228
9.1 Resistorpyramid(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . 230
9.2 Resistortetrahedron(cid:70) . . . . . . . . . . . . . . . . . . . . . 234
9.3 Resistorsquare(cid:70)(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . 235
9.4 Resistorcube(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . . 237
9.5 Powertransmission(cid:70) . . . . . . . . . . . . . . . . . . . . . 239
9.6 RMSpower(cid:70) . . . . . . . . . . . . . . . . . . . . . . . . . 240
9.7 Boilingtime(cid:70) . . . . . . . . . . . . . . . . . . . . . . . . . 240
10 Gravity 242
10.1 Thehollowmoon(cid:70) . . . . . . . . . . . . . . . . . . . . . . 243
10.2 Lowest-energycircularorbit(cid:70)(cid:70) . . . . . . . . . . . . . . . 245
10.3 Weightlessinspace(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . 247
10.4 Jumpintospace(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . 250
10.5 Spacegraveyard(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . 254
10.6 Newton’scannonball(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . 259
10.7 DelaTerreàlaLune(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . 263
10.8 ProfessorPlumb’sAstrolabe-Plumb(cid:70)(cid:70)(cid:70) . . . . . . . . . . 265
10.9 Jetaircraftdiet(cid:70)(cid:70). . . . . . . . . . . . . . . . . . . . . . . 269
10.10EscapevelocityfromtheSolarSystem(cid:70)(cid:70)(cid:70) . . . . . . . . . 272
10.11MrMegalopolis’expandingMoon(cid:70)(cid:70)(cid:70) . . . . . . . . . . . 276
10.12Asteroidgames(cid:70)(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . 279
11 Optics 285
11.1 Moteinasphere(cid:70) . . . . . . . . . . . . . . . . . . . . . . 285
11.2 Diminishingringsoflight(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . 286
11.3 Floatingpigs(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . . 288
11.4 TheMartianandthecaveman(cid:70)or(cid:70)(cid:70)(cid:70)(cid:70) . . . . . . . . . 293
11.5 Strangefish(cid:70)(cid:70)(cid:70)or(cid:70)(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . 298
12 Heat 305
12.1 Theheatedplate(cid:70) . . . . . . . . . . . . . . . . . . . . . . . 306
12.2 Theheatedcube(cid:70) . . . . . . . . . . . . . . . . . . . . . . . 308
12.3 Fridgeinaroom(cid:70)(cid:70). . . . . . . . . . . . . . . . . . . . . . 310
12.4 Iceinthedesert(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . 314
12.5 ThecoldendoftheEarth(cid:70)(cid:70). . . . . . . . . . . . . . . . . 318
x CONTENTS
13 Buoyancyandhydrostatics 322
13.1 Archimedes’crownandGalileo’sbalance(cid:70). . . . . . . . . . 323
13.2 AnotherGalileo’sbalancepuzzle(cid:70)(cid:70) . . . . . . . . . . . . . 328
13.3 Balancedscales(cid:70) . . . . . . . . . . . . . . . . . . . . . . . . 331
13.4 Thefloatingballandthesinkingball(cid:70)(cid:70) . . . . . . . . . . . 332
13.5 Floatingcylinders(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . 333
13.6 Thehydrostaticparadox(cid:70)(cid:70) . . . . . . . . . . . . . . . . . 336
13.7 Aquantitativepistonpuzzle(cid:70) . . . . . . . . . . . . . . . . 338
13.8 Thefloatingbar(cid:70)(cid:70)(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . 340
14 Estimation 351
14.1 Mile-hightower(cid:70) . . . . . . . . . . . . . . . . . . . . . . . 352
14.2 Howlongdowehaveleft? (cid:70)(cid:70) . . . . . . . . . . . . . . . . 355
14.3 Midas’storeroom(cid:70) . . . . . . . . . . . . . . . . . . . . . . 357
14.4 NapoleonBonaparteandtheGreatPyramid(cid:70) . . . . . . . 358
14.5 LawnchairLarry(cid:70)(cid:70) . . . . . . . . . . . . . . . . . . . . . 360
14.6 Dowegetlighterbybreathing? (cid:70)(cid:70) . . . . . . . . . . . . . 365
TheDeadlyGameofPuzzlePoints 367
Endnote 374
Introduction
ThisisapersonalbookwhichcelebratesapassionIhavealwayshadforplayful
problems in physics and maths. Questions that rely on material no harder
thanthatstudiedinhighschool,butwhichencourageoriginalthinking,and
stretchusinnewandinterestingdirections.
In practically every country that has an established higher education sys-
tem, the most competitive universities operate some kind of admissions test
orinterview. Thesegenerallyhaveonlyoneobject: todistinguishfrommany
brilliant applicants those with the greatest potential. To test this potential,
questions are often designed to test an applicant’s ability to think creatively.
In many countries, these pre-university tests are set by university professors.
Whilst the questions might reference a particular high-school syllabus, they
wouldnotnormallybeconstrainedbyit. Thequestionsareoftendeliberately
off-beat. Thisistoseeifapplicantscanapplywhattheyhavealreadylearntin
newandmorechallengingsituations. Thequestionstesttheapplicants’ability
to pick an unusual problem apart, reduce it to its essential components, and
usestandardtoolstosolveit.
Thisbookisacollectionofsomeofmyfavouritepre-universityproblems
inphysicsandmaths. Thesequestionsaredevisedtoencouragecuriosityand
playfulness,andmanyareofthestandardexpectedinsomeuniversityentrance
tests. These questions are perplexing, puzzling, but—most of all—fun. You
should regard them like toys. Pick up the one that most appeals, and play
with it. When you have exhausted it, you can entertain a friend with it. It
might seem impossibly nerdy, but for me almost nothing is as enjoyable as
beingbaffledbyanapparentlysimpleprobleminanareaofclassicalphysicsI
thoughtIunderstood. ThisbookisawayofsharingthepleasureIhavetaken
insomeofthephysicsandmathspuzzlesIhavemostenjoyed. Thequestions
will appeal to those high-school students who have mastered the basics, and
feeltheyhaveroomtoplayalittlewithsomethingmoreunusual.Teacherscan
alsousetheproblemstostretchtheirpupilswithsomethingmorechallenging
or unconventional. Here I am grateful to those teachers who allowed their
classes to pilot these problems, and to the students who bravely attempted
them.
I invented many of the questions myself. Others were suggested or in-
spired by friends and colleagues with an interest in this book. Still more are
1
