P 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