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Professor Stewart's Casebook of Mathematical Mysteries Professor Stewart’s Casebook of Mathematical Mysteries Professor Ian Stewart is known throughout the world for making mathematics popular. He received the Royal Society’s Faraday Medal for furthering the public understanding of science in1995, the IMA Gold Medal in 2000, the AAAS Public Understanding of Science and Technology Award in 2001 and the LMS/IMA Zeeman Medal in 2008. He was elected a Fellow of the Royal Society in 2001. He is Emeritus Professor of Mathematics at the University of Warwick, where he divides his time between research into nonlinear dynamics and furthering public awareness of mathematics. His many popular science books include (with Terry Pratchett and Jack Cohen) The Science of Discworld I to IV, The Mathematics of Life, 17 Equations that Changed the World and The Great Mathematical Problems. His app, Professor Stewart’s Incredible Numbers, was published jointly by Profile and Touch Press in March 2014. By the Same Author Concepts Of Modern Mathematics Game, Set, And Math Does God Play Dice? Another Fine Math You’ve Got Me Into Fearful Symmetry Nature’s Numbers From Here To Infinity The Magical Maze Life’s Other Secret Flatterland What Shape Is A Snowflake? The Annotated Flatland Math Hysteria The Mayor Of Uglyville’s Dilemma How To Cut A Cake Letters To A Young Mathematician Taming The Infinite (Alternative Title: The Story Of Mathematics) Why Beauty Is Truth Cows In The Maze Mathematics Of Life Professor Stewart’s Cabinet Of Mathematical Curiosities Professor Stewart’s Hoard Of Mathematical Treasures Seventeen Equations That Changed The World (Alternative Title: In Pursuit Of The Unknown) The Great Mathematical Problems (Alternative Title: Visions Of Infinity) Symmetry: A Very Short Introduction Jack Of All Trades (Science Fiction eBook) with Jack Cohen The Collapse Of Chaos Evolving The Alien (Alternative Title: What Does A Martian Look Like?) Figments Of Reality Wheelers (Science Fiction) Heaven (Science Fiction) The Science Of Discworld Series (With Terry Pratchett & Jack Cohen) The Science Of Discworld The Science Of Discworld II: The Globe The Science Of Discworld III: Darwin’s Watch The Science Of Discworld IV: Judgement Day iPad app Incredible Numbers Professor Stewart’s Casebook of Mathematical Mysteries Ian Stewart First published in 2014 by PROFILE BOOKS LTD 3A Exmouth House Pine Street London EC1R 0JH www.profilebooks.com Copyright © Joat Enterprises 2014 10 9 8 7 6 5 4 3 2 1 Text design by Sue Lamble Typeset in Stone Serif by Data Standards Ltd, Frome, Somerset. The moral right of the author has been asserted. All rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior written permission of both the copyright owner and the publisher of this book. A CIP catalogue record for this book is available from the British Library. eISBN 978 1 84765 432 8 CONTENTS Acknowledgements Introducing Soames and Watsup Note on Units The Scandal of the Stolen Sovereign Number Curiosity Track Position Soames Meets Watsup Geomagic Squares What Shape is an Orange Peel? How to Win the Lottery? The Green Socks Caper Incident Consecutive Cubes Adonis Asteroid Mousterian Two Square Quickies Caught Clean-Handed The Adventure of the Cardboard Boxes The RATS Sequence Birthdays Are Good for You Mathematical Dates The Hound of the Basketballs Digital Cubes Narcissistic Numbers Piphilology, Piems, and Pilish Clueless! A Brief History of Sudoku Hexakosioihexekontahexaphobia Once, Twice, Thrice Conservation of Luck The Case of the Face-Down Aces Confused Parents Jigsaw Paradox The Catflap of Fear Pancake Numbers The Soup Plate Trick Mathematical Haiku The Case of the Cryptic Cartwheel Two by Two The V-shaped Goose Mystery Eelish Mnemonics Amazing Squares The Thirty-Seven Mystery Average Speed Four Clueless Pseudoku Sums of Cubes The Puzzle of the Purloined Papers Master of All He Surveys Another Number Curiosity The Opaque Square Problem Opaque Polygons and Circles πr2? The Sign of One Progress on Prime Gaps The Odd Goldbach Conjecture Prime Number Mysteries The Optimal Pyramid The Sign of One: Part Two Initial Confusion Euclid’s Doodle Euclidean Efficiency 123456789 Times X The Sign of One: Part Three Taxicab Numbers The Wave of Translation Riddle of the Sands Eskimo π The Sign of One: Part Four—Concluded Seriously Deranged Tossing a Fair Coin Isn’t Fair Playing Poker by Post Eliminating the Impossible Mussel Power Proof That the World is Round 123456789 Times X Continued The Price of Fame The Riddle of the Golden Rhombus A Powerful Arithmetic Sequence Why Do Guinness Bubbles Go Downwards? Random Harmonic Series The Dogs That Fight in the Park How Tall is That Tree? Why Do My Friends Have More Friends Than I Do? Isn’t Statistics Wonderful? The Adventure of the Six Guests How to Write Very Big Numbers Graham’s Number Can’t Wrap My Head Around It The Affair of the Above-Average Driver The Mousetrap Cube Sierpiński Numbers James Joseph Who? The Baffleham Burglary The Quadrillionth Digit of π Is π Normal? A Mathematician, a Statistician, and an Engineer… Lakes of Wada Fermat’s Last Limerick Malfatti’s Mistake Square Leftovers Coin Tossing over the Phone How to Stop Unwanted Echoes The Enigma of the Versatile Tile The Thrackle Conjecture Bargain with the Devil A Tiling That Is Not Periodic The Two Colour Theorem The Four Colour Theorem in Space Comical Calculus The Erdős Discrepancy Problem The Greek Integrator Sums of Four Cubes Why the Leopard Got Its Spots Polygons Forever Top Secret The Adventure of the Rowing Men The Fifteen Puzzle The Tricky Six Puzzle As Difficult as ABC Rings of Regular Solids The Square Peg Problem The Impossible Route The Final Problem The Return The Final Solution The Mysteries Demystified Acknowledgements page 13 Left and centre figures: Laurent Bartholdi and André Henriques. Orange peels and Fresnel integrals, Mathematical Intelligencer 34 No. 4 (2012) 1–3. page 13 Right figure: Luc Devroye. page 27 Box puzzle concept: Moloy De. page 40 Extract from Not a Wake: Mike Keith. page 62 Haiku: Daniel Mathews, Jonathan Alperin, Jonathan Rosenberg. page 67 Figure: http://getyournotes.blogspot.co.uk/2011/08/why-do-some-birds- fly-in-v-formations.html page 71 Amazing squares: devised by Moloy De and Nirmalya Chattopadhyay. page 72 The Thirty-Seven Mystery: based on observations by Stephen Gledhill. page 74 Clueless pseudoku: Gerard Butters, Frederick Henle, James Henle and Colleen McGaughey. Creating clueless puzzles, The Mathematical Intelligencer 33 No. 3 (Fall 2011) 102–105. page 96 Figure: Eric W. Weisstein, ‘Brocard’s Conjecture,’ from MathWorld— A Wolfram Web Resource: http://mathworld.wolfram.com/BrocardsConjecture.html page 101 Right Figure: Steven Snape. page 109 Figure: Courtesy of the UW-Madison Archives. page 123 Left figure: [George Steinmetz, courtesy of Anastasia Photo]. page 123 Right figure: NASA, HiRISE on Mars Reconnaissance Orbiter. page 124 Right figure: Rudi Podgornik. page 125 Figure: Veit Schwa¨mmle and Hans J. Herrmann. Solitary wave behaviour of sand dunes, Nature 426 (2003) 619–620. page 131 Figure: Persi Diaconis, Susan Holmes and Richard Montgomery, Dynamical bias in the coin toss, SIAM Review 49 (2007) 211–223. page 205 Figures: Joshua Socolar and Joan Taylor. An aperiodic hexagonal tile, Journal of Combinatorial Theory Series A 118 (2011) 2207–2231; http://link.springer.com/article/10.1007%2Fs00283-011-9255-y pages 235–7 Figures: Michael Elgersma and Stan Wagon, Closing a Platonic gap, The Mathematical Intelligencer (2014) to appear. The following figures are reproduced under the Creative Commons Attribution 3.0 Unported license and credited as requested on the description page: page 95 Krishnavedala. page 101 (left) Ricardo Liberato. page 102 Tekisch. page 139 Andreas Trepte, www.photo-natur.de. page 180 Braindrain0000. page 183 LutzL. page 296 Walters Art Museum, Baltimore. Introducing Soames and Watsup Professor Stewart’s Cabinet of Mathematical Curiosities appeared in 2008, just before Christmas. Readers seemed to like its random mixture of quirky mathematical tricks, games, weird biographies, snippets of strange information, solved and unsolved problems, odd factoids, and the occasional longer and more serious piece on topics such as fractals, topology, and Fermat’s Last Theorem. So in 2009 it was followed by Professor Stewart’s Hoard of Mathematical Treasures, which continued in the same vein with an intermittent pirate theme. They say that three is a good number for a trilogy. The late Douglas Adams of The Hitchhiker’s Guide to the Galaxy fame did eventually decide that four was better and five better still, but three sounds like a good place to start. So, after a gap of five years, here is Professor Stewart’s Casebook of Mathematical Mysteries. This time, however, there’s a new twist. The short quirky items, such as Hexakosioihexekontahexaphobia, the Thrackle Conjecture, What Shape is an Orange Peel?, the RATS Sequence, and Euclid’s Doodle, are still there. So are more substantial articles about solved and unsolved problems: Pancake Numbers, the Goldbach Conjecture, the Erdős Discrepancy Problem, the Square Peg Conjecture, and the ABC Conjecture. So are the jokes, poems, and anecdotes. Not to mention unusual applications of mathematics to flying geese, clumps of mussels, spotty leopards, and bubbles in Guinness. But these miscellanea are now interspersed with a series of narrative episodes featuring a Victorian detective and his medical sidekick— I know what you’re thinking. However, I developed the idea a year or so before Benedict Cumberbatch and Martin Freeman’s spectacularly successful modern take on Sir Arthur Conan Doyle’s much-loved characters hit the small screen. (Trust me.) More to the point, it’s not that pair.
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