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THE MATHEMATICS OF OZ

Mental Gymnastics from Beyond the Edge

Grab a pencil. Relax. Then take off on a mind-boggling journey to theultimate frontier of math, mind, and meaning as acclaimed author Dr.Clifford Pickover, Dorothy, and Dr. Oz explore some of the oddest andquirkiest highways and byways of the numerically obsessed. Prepareyourself for a shattering odyssey as The Mathematics of Oz unlocks thedoors of your imagination. The thought-provoking mysteries, puzzles,and problems range from zebra numbers and circular primes to Legion’snumber – a number so big that it makes a trillion pale in comparison. Thestrange mazes, bizarre consequences, and dizzying arrays of logic prob-lems will entertain people at all levels of mathematical sophistication.

The tests devised by enigmatic Dr. Oz to assess human intelligencewill tease the brain of even the most avid puzzle fan. Test your wits on ahost of mathematical topics: geometry and mazes, sequences, series, sets,arrangements, probability and misdirection, number theory, arithmetic,and even several problems dealing with the physical world. With numer-ous illustrations, this is an original, fun-filled, and thoroughly uniqueintroduction to numbers and their role in creativity, computers, games,practical research, and absurd adventures that teeter on the edge of logicand insanity. The Mathematics of Oz will have you squirming in frustra-tion and begging for more.

Clifford A. Pickover received his Ph.D. from Yale University and is theauthor of over thirty highly-acclaimed books on topics such as comput-ers and creativity, art, mathematics, black holes, human behavior andintelligence, time travel, alien life, and science fiction. Pickover is a pro-lific inventor with dozens of patents, is the associate editor for severaljournals, the author of colorful puzzle calendars, and puzzle contributorto magazines geared to children and adults. The Los Angeles Times wrote,“Pickover has published nearly a book a year in which he stretches thelimits of computers, art, and thought.” Wired magazine wrote, “BuckyFuller thought big, Arthur C. Clarke thinks big, but Cliff Pickover out-does them both.” Pickover’s computer graphics have been featured onthe cover of many popular magazines and on TV shows. His web site,www.pickover.com, has received over a million visits.

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Works by Clifford A. Pickover

The Alien IQ TestBlack Holes: A Traveler’s GuideChaos and FractalsChaos in WonderlandComputers, Pattern, Chaos, and BeautyCryptorunes: Codes and Secret WritingDreaming the FutureFuture Health: Computers and Medicine in the 21st CenturyFractal Horizons: The Future Use of FractalsFrontiers of Scientific Visualization (with Stu Tewksbury)The Girl Who Gave Birth to RabbitsKeys to InfinityThe Loom of GodMazes for the Mind: Computers and the UnexpectedMind-Bending Visual Puzzles (calendars and card sets)The Paradox of God and the Science of OmniscienceThe Pattern Book: Fractals, Art, and NatureThe Science of AliensSpider Legs (with Piers Anthony)Spiral Symmetry (with Istvan Hargittai)Strange Brains and GeniusThe Stars of HeavenSurfing through HyperspaceTime: A Traveler’s GuideVisions of the FutureVisualizing Biological InformationWonders of NumbersThe Zen of Magic Squares, Circles, and Stars

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THEMATHEMATICS OFOZMental Gymnastics fromBeyond the Edge

CLIFFORD A. PICKOVER

Cambridge University Press978-0-521-70084-9 - The Mathematics of OZ: Mental Gymnastics from Beyond the EdgeClifford A. PickoverFrontmatterMore information

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cambridge university pressCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press32 Avenue of the Americas, New York, ny 10013 -2473 , usa

www.cambridge.orgInformation on this title: www.cambridge.org/9780521016780

C© Clifford A. Pickover 2002

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.

First published 2002First paperback edition 2007

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication Data

Pickover, Clifford A.The mathematics of Oz: mental gymnastics from beyond the edge / Clifford A. Pickover.

p. cm.Includes bibliographical references and index.isbn 0-521-01678-91. Mathematical recreations. I. Title.qa95 .p528 2002793 .7′4–dc21 2002023346

isbn 978-0-521-01678-0 hardbackisbn 978-0-521-70084-9 paperback

Cambridge University Press has no responsibility forthe persistence or accuracy of urls for external orthird-party Internet Web sites referred to in this publicationand does not guarantee that any content on suchWeb sites is, or will remain, accurate or appropriate.

Cambridge University Press978-0-521-70084-9 - The Mathematics of OZ: Mental Gymnastics from Beyond the EdgeClifford A. PickoverFrontmatterMore information

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This book is dedicated to my uncle, Dr. Bruce Pickover,

who stimulated my early interest in mathematical and other puzzles.

A good glass in the bishop's

hostel in the devil's seat

61 Sphere62 Potawatomi Target63 Sliders64 Swapping65 Triangle Dissection66 A Simple Code67 Heterosquares68 Insertion69 Missing Landscape70 The Choice71 Animal Selection72 The Skeletal Men of Uranus73 Hindbrain Stimulation74 The Arrays of Absolution75 Trochophore Abduction76 The Dream Pyramids of Missouri77 Mathematical Flower Petal78 Blood and Water79 Cavern Problem80 Three Triplets81 Oos and Oob Gambit82 Napiform Mathematics83 Toto, Mr. Plex, Elephant84 Witch Overdrive85 What Is Art?86 Wendy Magic Square87 Heaven and Hell88 The Stars of Heaven89 Vacation in the Tarantula Nebula90 Hot Lava91 Circular Primes92 The Truth about Cats and Dogs93 Disc Mania94 n2 + m2 = s95 2, 271, 271828196 Android Watch97 More Knight Moves98 Pool Table Gambit99 A Connection between π and e

100 Venusian Number Bush101 Triangle Cave

Contents

140141142144146148149151153155156158160161162163165166167169171173174176180182184186188189190192194196198199201204206209211

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102 Rat Attack103 The Scarecrow Formula104 Circle Mathematics105 A, AB, ABA106 Ants and Cheese107 The Omega Crystal108 Attack of the Undulating Undecamorphs

Epilogue

Further ExploringFurther ReadingIndex

x Contents

214217219221223224226

231

235347349

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He calmly rode on, leaving it to his horse's discretion to go which way it pleased,firmly believing that in this consisted the very essence of adventures.

— Miguel de Cervantes, Don Quixote

The road through this book is chaotic and takes many turns in order tosurprise and delight you. However, if you wish to take your hovercraftand jump between mathematical puzzles of similar kinds, the followingguide should help.

HACEFGHIJKMBHÄ You are here➥ Geometry (Chapters 1, 3, 8, 14, 16, 17, 18, 23, 47, 50, 54, 55, 58,

61, 65, 84, 88, 96, 103, 104, 106)➥ Mazelike Problems (Chapters 6, 12, 13, 15, 22, 24, 36, 46, 49, 52,

60, 83, 87, 97, 98, 101, 102) ➥ Sequences, Series, Sets, and Arrangements (Chapters 2, 4, 5, 9,

11, 25, 26, 30, 34, 37, 38, 41, 48, 50, 56, 59, 63, 64, 66, 69, 71, 72,73, 74, 79, 80, 82, 85, 86, 89, 92, 93, 107)

➥ Physical World (Chapters 1, 3, 7, 40, 44, 45, 78, 102)➥ Probability and Misdirection (Chapters 8, 14, 17, 19, 27, 31, 32,

51, 70, 81, 90)➥ Number Theory and Arithmetic (Chapters 9, 10, 20, 21, 27, 33,

35, 37, 39, 42, 43, 45, 47, 53, 57, 62, 67, 68, 76, 77, 82, 91, 99, 100,105, 108)End here ☞ÄB(Freedom for Dorothy and Ultimate Reader Enlightenment)

☞

Travel Guide

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xiii

“If you only had brains in your head you would be as good a man as any of them,and a better man than some of them. Brains are the only things worth having inthis world, no matter whether one is a crow or a man.”

— Crows talking to the Scarecrow in The Wonderful Wizard of Oz

Oz is a metaphor for mystery. Oz is a state of mind. Oz is a parallel uni-verse that may exist side by side with our own in some ghostly fashion.

Published in 1900 by L. Frank Baum, The Wonderful Wizard of Ozstarred young Dorothy of Kansas along with a magical array of charac-ters ranging from a Scarecrow and a Tin Woodman to a phony Wizardwho used magic to help Dorothy come to realize that there is no placelike home.

In The Mathematics of Oz Dorothy is certainly far from home. Abduct-ed by mathematically obsessed aliens, Dorothy tries to solve a bafflingarray of brainteasers that often center around numbers and mathematics.The aliens’ obsession with mathematics probably sounds silly to manyof you, but numerical challenges are a great way to transcend space andtime. Mathematics is a universal language, and numbers might be ourfirst means of communication with intelligent alien races.

Dorothy, Dr. Oz (her abductor), and Mr. Plex (Oz’s assistant) have a lim-ited attention span, and they don’t want you readers to wade throughpages of background before getting to the essential ingredients. To avoidthis problem, each chapter in this book is just a few pages in length. Oneadvantage of this format is that you can jump right in to experiment andhave fun. The book is not intended for mathematicians looking for for-mal mathematical explanations; however, additional material can befound in the “Further Exploring” and “Further Reading” sections.

Prepare yourself for a strange journey as The Mathematics of Oz un-locks the doors of your imagination. The mysteries, puzzles, and prob-

Preface

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lems range from building a yellow-brick road that crosses America, tozebra numbers and circular primes, to Legion’s number – a number sobig that it makes a trillion pale in comparison – to “The Problem of theBones,” a fiendishly difficult mathematical problem involving probabil-ity and shattering of leg bones.

Grab a pencil. Do not fear. Some of the topics in the book may appearto be curiosities, with little practical application or purpose. However, Ihave found these experiments to be useful and educational – as havethe many students, educators, and scientists who have written to me.Throughout history, experiments, ideas and conclusions originating inthe play of the mind have found striking and unexpected practical ap-plications. Or as mathematician Gottfried Wilhelm Leibniz once said, “Leshommes ne sont jamais plus ingénieux que dans l’invention des jeux.”(“Men are never more ingenious than when they are inventing games.”)

❈ ❈ ❈

This book is for anyone who wants to enter new mental worlds. If youare a teacher, you may want to use the mathematical brain teasers tostimulate students. Have them design their own puzzles similar to theones in this book. Computer programmers may want to create or solvesimilar puzzles using a computer, although a computer is definitely notnecessary to attack and solve the problems in this book.

To help you assess your level of performance during your journeythrough this book, I have assigned difficulty ratings to the various puz-zles:

7 Challenging7 7 Very challenging7 7 7 Extremely difficult7 7 7 7 Outrageously difficult: probably impossible for Dorothy and

other hom*o sapiens to solve.

To retain the playful spirit of the book, and its sense of crazy adven-ture, puzzles with different difficulty levels are scattered randomlythrough the book – as if the puzzles had been tossed about by a tor-nado. Browse from the mathematical smorgasbord and feed your mind.

xiv Preface

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Every now and again one comes across an astounding result that closely relatestwo foreign objects which seem to have nothing in common. Who would suspect,for example, that on the average, the number of ways of expressing a positiveinteger n as a sum of two integral squares, x2 + y2 = n, is π.

— Ross Honsberger, Mathematical Gems III

I thank Brian C. Mansfield for his wonderful cartoon diagrams, usedthroughout the book. Over the years, Brian has been helpful beyondcompare.

Numerous people have provided very useful feedback and informationfor the solutions to my puzzles: Dennis Gordon, Robert Stong, David T.Blackston, Dennis Yelle, Balakumar Jothimohan Balasubramaniam, IlanMayer, Ed Murphy, Jim Gillogly, Dan Tilque, Bill Ryan, James Van Buskirk,“R.E.S.,” Dennis Gordon, Dharmashankar Subramanian, Richard Heath-field, Al Zimmerman, Risto Lankinen, Seth Breidbart, Darrell Plank, DavidA. Karr, Jason Earls, Ken Inoue, and others.

I thank Samuel Marcius for the symbol L, which represents Mr. Plex,and for other alienlike symbols used in this book. The animal font is a freeware font by Alan Carr. Ann Stretton designed the font that con-tains symbols such as 7. Symbols like u are part of freeware from OmegaFont Labs. Michael Lee and Josh Dixon designed the font that looks likeTHIS.

All Oz quotations come from L. Frank Baum’s classic novels, The Wiz-ard of Oz (originally published in 1900 as The Wonderful Wizard of Oz),The Land of Oz (originally published in 1904 as The Marvelous Land ofOz), Ozma of Oz (1907), Dorothy and the Wizard in Oz (1908), The Roadto Oz (1909), The Emerald City of Oz (1910), The Patchwork Girl of Oz(1913), Tik-Tok of Oz (1914), The Scarecrow of Oz (1915), Rinkitink in Oz(1916), The Lost Princess of Oz (1917), The Tin Woodman of Oz (1918), The

Acknowledgments

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Magic of Oz (1919), and Glinda of Oz (1920). For more information onOz, see Eric P. Gjovaag’s “Oz” Web site: http://www.eskimo.com/~tiktok/.

Note: As many readers are aware, Internet Web sites come and go.Sometimes they change addresses or completely disappear. The Web siteaddresses (URLs) listed in this book provided valuable background in-formation when this book was written. You can, of course, find numerousother sites relating to many of the mathematical puzzles and topics inthis book by using search engines such as www.google.com.

xvi Acknowledgments

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References