ffirs.qxd 5/13/05 11:21 AM Page iii AA PassionPassion forfor MathematicsMathematics Numbers, Puzzles, Madness, Religion, and the Quest for Reality CLIFFORD A. PICKOVER John Wiley & Sons, Inc. ffirs.qxd 5/13/05 11:21 AM Page vi ffirs.qxd 5/13/05 11:21 AM Page i AA PassionPassion forfor MathematicsMathematics ffirs.qxd 5/13/05 11:21 AM Page ii Works by Clifford A. Pickover The Alien IQ Test Black Holes: A Traveler’s Guide Calculus and Pizza Chaos and Fractals Chaos in Wonderland Computers, Pattern, Chaos, and Beauty Computers and the Imagination Cryptorunes: Codes and Secret Writing Dreaming the Future Egg Drop Soup Future Health Fractal Horizons: The Future Use of Fractals Frontiers of Scientific Visualization The Girl Who Gave Birth to Rabbits Keys to Infinity Liquid Earth The Lobotomy Club The Loom of God The Mathematics of Oz Mazes for the Mind: Computers and the Unexpected Mind-Bending Visual Puzzles (calendars and card sets) The Paradox of God and the Science of Omniscience The Pattern Book: Fractals, Art, and Nature The Science of Aliens Sex, Drugs, Einstein, and Elves Spider Legs (with Piers Anthony) Spiral Symmetry (with Istvan Hargittai) Strange Brains and Genius Sushi Never Sleeps The Stars of Heaven Surfing through Hyperspace Time: A Traveler’s Guide Visions of the Future Visualizing Biological Information Wonders of Numbers The Zen of Magic Squares, Circles, and Stars ffirs.qxd 5/13/05 11:21 AM Page iii AA PassionPassion forfor MathematicsMathematics Numbers, Puzzles, Madness, Religion, and the Quest for Reality CLIFFORD A. PICKOVER John Wiley & Sons, Inc. ffirs.qxd 5/13/05 11:21 AM Page iv Copyright © 2005 by Clifford A. Pickover. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada Illustration credits: pages 91, 116, 137, 140, 142, 149, 150, 151, 157, 158, 159, 160, 162, 164, 167, 168, 169, 179, 199, 214, 215, 224, 225, 230, 274, 302, 336, 338, 341, 343, 345, and 348 by Brian C. Mans- field; 113, 114, 115, 145, 146, 332, 333, and 334 by Sam Loyd; 139 courtesy of Peter Hamburger and Edit Hepp; 141 by Stewart Raphael, Audrey Raphael, and Richard King; 155 by Patrick Grimm and Paul St. Denis; 165 and 166 from Magic Squares and Cubes by W. S. Andrews; 177 and 352 by Henry Ernest Dudeney; 200 by Bruce Patterson; 204 by Bruce Rawles; 206 by Jürgen Schmidhuber; 253 by Abram Hindle; 254, 255, 256, and 257 by Chris Coyne; 258 and 259 by Jock Cooper; 260 by Linda Bucklin; 261 by Sally Hunter; 262 and 263 by Jos Leys; 264 by Robert A. Johnston; and 266 by Cory and Catska Ench. Design and composition by Navta Associates, Inc. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permit- ted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646- 8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or com- pleteness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, inci- dental, consequential, or other damages. For general information about our other products and services, please contact our Customer Care Depart- ment within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Pickover, Clifford A. A passion for mathematics : numbers, puzzles, madness, religion, and the quest for reality / Clifford A. Pickover. p. cm. Includes bibliographical references and index. ISBN-13 978-0-471-69098-6 (paper) ISBN-10 0-471-69098-8 (paper) 1. Mathematics. I. Title QA39.3.P53 2005 510—dc22 2004060622 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ffirs.qxd 5/13/05 11:21 AM Page v Ramanujan said that he received his formulas from God. This book is dedicated to all those who find Ramanujan’s π formulas beautiful to look at: 1 ∞ ()()!!1103+−− 26390nn 2 1()!! 4 n 1 = 8 ∑ π 42nn+ 3 n =0 99 32 (!)n n 0 () −1 1 1 5 ∞ ()11n + 1 ( ) ( ) ( ) n 55 2 Poch() n6 Poch () n6 Poch () n 4 π = ∑ 3 23 n =0 (!)n 125 (where Poch(n) refers to the Pochhammer notation described in chapter 2) ffirs.qxd 5/13/05 11:21 AM Page vi ffirs.qxd 5/13/05 11:21 AM Page vii “Truly the gods have not from the beginning revealed all things to mortals, but by long seeking, mortals make progress in discovery.” —Xenophanes of Colophon (c. 500 B.C.) “Every blade of grass has its angel that bends over it and whispers, ‘grow, grow.’” —Talmudic commentary Midrash Bereishis Rabbah, 10:6 ffirs.qxd 5/13/05 11:21 AM Page viii ftoc.qxd 5/17/05 9:37 AM Page ix Answers ContentsContents Acknowledgments Introduction Acknowledgments xiii Introduction 1 1. Numbers, History, Society, and People 11 In which we encounter religious mathematicians, mad mathematicians, famous mathematicians, mathematical savants, quirky questions, fun trivia, brief biographies, mathematical gods, historical oddities, numbers and society, gossip, the history of mathematical notation, the genesis of numbers, and “What if?” questions. 2. Cool Numbers 45 In which we encounter fascinating numbers and strange number sequences. We’ll explore transcendental numbers, octonions, surreal numbers, obstinate numbers, cyclic numbers, Vibonacci numbers, perfect numbers, automorphic numbers, prime numbers, Wilson primes, palindromic primes, Fibonacci primes, Sophie Germain primes, Baxter-Hickerson primes, star-congruent primes, narcissistic numbers, amenable numbers, amicable numbers, p-adic numbers, large palindromes, factorions, hyperfactorials, primorials, ix ftoc.qxd 5/17/05 9:37 AM Page x palindions and hyperpalindions, exotic-looking formulas for π, the Golay- Rudin-Shapiro sequence, the wonderful Pochhammer notation, and famous and curious math constants (such as Liouville’s constant, the Copeland- Erdös constant, Brun’s constant, Champernowne’s number, Euler’s gamma, Chaitin’s constant, the Landau-Ramanujan constant, Mills’s constant, the golden ratio, Apéry’s constant, and constants even more bizarre). 3. Algebra, Percentages, Weird Puzzles, and Marvelous Mathematical Manipulations 111 In which we encounter treasure chests of zany and educational math problems that involve algebra, fractions, percentages, classic recreational puzzles, and various types of mathematical manipulation. Some are based on problems that are more than a thousand years old. Others are brand new. Get ready to sharpen your pencils and stretch your brains! 4. Geometry, Games, and Beyond 135 In which we explore tiles, patterns, position problems, arrays, Venn dia- grams, tic-tac-toe, other games played on boards, Königsberg bridges, catenaries, Loyd’s and Dudeney’s puzzles, Sherck’s surface, magic squares, lituuses, inside-out Mandelbrot sets, the Quadratrix of Hippias, hyper- spheres, fractal geese, Schmidhuber circles, Pappus’s Arbelos, Escher patterns, and chess knights. 5. Probability: Take Your Chances 209 In which we explore casinos, logic, guessing, decisions, combinations, permutations, competition, possibilities, games involving choice, monkeys typing Hamlet, Benford’s law, combinatorics, alien gambits, Rubik’s Cubes, card shuffles, marble mazes, nontransitive dice, dangerous movements of air molecules, the board game of the gods, and the tunnels of death and despair. x Contents ftoc.qxd 5/17/05 9:37 AM Page xi 6. Big Numbers and Infinity 233 In which we explore very large numbers, the edges of comprehension, infin- ity, the Funnel of Zeus, the infinite gift, the omega crystal, the Skewes num- ber, the Monster group, Göbel’s number, the awesome quattuordecillion, the Erdös-Moser number, Archimedes’ famous “Cattle Problem,” classic para- doxes, Gauss’s “measurable infinity,” and Knuth’s arrow notation. 7. Mathematics and Beauty 251 In which we explore artistic forms generated from mathematics—delicate fungi, twenty-second-century cityscapes, fractal necklaces and seed pods, alien devices, and a rich panoply of patterns that exhibit a cascade of detail with increasing magnifications. Answers 267 References 383 Index 387 Contents xi ftoc.qxd 5/17/05 9:37 AM Page xii Answers flast.qxd 5/13/05 11:00 AM Page xiii Contents AcknowledgmentsAcknowledgments Introduction I thank Brian Mansfield for his wonderful cartoon diagrams that appear throughout the book. Over the years, Brian has been helpful beyond compare. Numerous people have provided useful feedback and information relating to the solutions to my puzzles; these individuals include Dennis Gordon, Robert Stong, Paul Moskowitz, Joseph Pe, Daniel Dockery, Mark Nandor, Mark Ganson, Nick Hobson, Chuck Gaydos, Graham Cleverley, Jeffrey Carr, Jon Anderson, “Jaymz” James Salter, Chris Meyers, Pete Barnes, Steve Brazzell, Steve Blattnig, Edith Rudy, Eric Baise, Martie Saxenmeyer, Bob Ewell, Teja Krasek, and many more.