I 1 1 1 ft 4 1 1 SOLVIN G CRIME WITH MATHEMATICS - * THE NUMBERS BEHIND NUMB3RS KEITH DEVLIN . NPR'S "Moth Guy" and GAR! ' LORDEhI , the M oth Consultant on NU MB3RS", the h it CBS tel evision series A COMPANION TO THE HIT CBS CRIME SERIES NUMB3RS PRESENTS THE FASCINATING WAYS MATHEMATICS IS USED TO FIGHT REAL-LIFE CRIME • :i k im Using the popular CBS prime-time TV crime series NUMB3RS' as a springboard, Keith Devlin (known to millions of NPR listeners as "the Math Guy" on NPR's Weekend Edition with Scott Simon) and Gary Lorden (the math consultant to NUMB3RS ") explain real-life mathematical techniques used by the FBI and other law enforcement agencies to catch and convict criminals. From forensics to counterterrorism. the Riemann hypothesis lo image enhancement, solving murders to beating casino odds, Devlin and Lorden present compelling cases that illustrate how ad­ vanced mathematics can be used in state-of-the-art criminal investigations. Praise for the television series: "NUMB3RS LOOKS LIKE A WINN3R." —USA Today A PLUME BOOK THE NUMBERS BEHIND NUMB3RS DR. KEITH DEVLIN is executive director of Stanford University's Center for the Study of Language and Information and a consulting professor of mathematics at Stanford. Devlin has a B.Sc. degree in Mathematics from King's College London (1968) and a Ph.D. in Mathematics from the Uni­ versity of Bristol (1971). He is a fellow of the American Association for the Advancement of Science, a World Economic Forum fellow, and a former member of the Mathematical Sciences Education Board of the U.S. National Academy of Sciences. The author of twenty-five books, Devlin has been a regular contributor to National Public Radio's popular program Weekend Edition, where he is known as "the Math Guy" in his on-air conversations with host Scott Simon. His monthly column, "Dev­ lin's Angle," appears on Mathematical Association of America's web journal MAA Online. DR. GARY LORDEN is a professor in the mathematics department of the California Institute of Technology in Pasadena. He graduated from Caltech with a B.S. in mathematics in 1962, received his Ph.D. in math­ ematics from Cornell University in 1966, and taught at Northwestern University before returning to Caltech in 1968. A fellow of the Institute of Mathematical Statistics, Lorden has taught statistics, probability, and other mathematics at all levels from freshman to doctoral. Lorden has also been active as a consultant and expert witness in mathematics and statistics for government agencies and laboratories, private companies, and law firms. For many years he consulted for Caltech's Jet Propulsion Laboratory for their space exploration programs. He has participated in highly classified research projects aimed at enhancing the ability of gov­ ernment agencies (such as the NSA) to protect national security. Lorden is the chief mathematics consultant for the CBS TV series NUMB3RS. THE NUMBERS BEHIND NUMB3RS Solving Crime with Mathematics Keith Devlin, Ph.D. and Gary Lorden, Ph.D. © A PLUME BOOK PLUME Published by Penguin Group Penguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, U.S.A. Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario, Canada M4P 2Y3 (a division of Pearson Penguin Canada Inc.) Penguin Books Ltd., 80 Strand, London WC2R 0RL, England Penguin Ireland, 25 St. Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd.) Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty. Ltd.) Penguin Books India Pvt. Ltd., 11 Community Centre, Panchsheel Park, New Delhi - 110 017, India Penguin Books (NZ), 67 Apollo Drive, Rosedale, North Shore 0745, Auckland, New Zealand (a division of Pearson New Zealand Ltd.) Penguin Books (South Africa) (Pty.) Ltd., 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa Penguin Books Ltd., Registered Offices: 80 Strand, London WC2R 0RL, England First published by Plume, a member of Penguin Group (USA) Inc. First Printing, September 2007 10 987654321 Copyright © Keith Devlin and Gary Lorden, 2007 All rights reserved Illustration credits appear on page 244. REGISTERED TRADEMARK—MARCA REGISTRADA LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Devlin, Keith J. The numbers behind NUMB3RS: solving crime with mathematics/Keith Devlin, Gary Lorden. p. cm. ISBN 978-0-452-28857-7 1. Criminal investigation. 2. Mathematical statistics. 3. Criminal investigation—Data processing. I. Title: Numbers behind numbers. II. Lorden, Gary. HI. Title. HV8073.5.D485 2007 363.2501'5195—dc22 2007018115 Printed in the United States of America Set in Dante MT Designed by Joseph Rutt Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in 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 above publisher of this book. PUBLISHER'S NOTE The scanning, uploading, and distribution of this book via the Internet or via any other means without the permission of the publisher is illegal and punishable by law. Please purchase only authorized electronic editions, and do not participate in or encour­ age electronic piracy of copyrighted materials. Your support of the author's rights is appreciated. BOOKS ARE AVAILABLE AT QUANTITY DISCOUNTS WHEN USED TO PROMOTE PRODUCTS OR SERVICES. FOR INFORMATION PLEASE WRITE TO PREMIUM MARKETING DIVISION, PENGUIN GROUP (USA) INC., 375 HUDSON STREET, NEW YORK, NEW YORK 10014. Acknowledgments The authors want to thank NUMB3RS creators Cheryl Heuton and Nick Falacci for creating Charlie Eppes, television's first mathematics super­ hero, and succeeding brilliantly in putting math on television in prime time. Their efforts have been joined by a stellar team of other writers, actors, producers, directors, and specialists whose work has inspired us to write this book. The gifted actor David Krumholtz has earned the undy­ ing love of mathematicians everywhere for bringing Charlie to life in a way that has led millions of people to see mathematics in a completely new light. Thanks also to NUMB3RS researchers Andy Black and Matt Kolokoff for being wonderful to work with in coming up with endless applications of mathematics to make the writers' dreams come true. We wish to express our particular thanks to mathematician Dr. Lenny Rudin of Cognitech, one of the world's foremost experts on im­ age enhancement, for considerable help with Chapter 5 and for provid­ ing the images we show in that chapter. Finally, Ted Weinstein, our agent, found us an excellent publisher in David Cashion of Plume, and both worked tirelessly to turn a manuscript that we felt was as reader-friendly as possible, given that this is a math book, into one that, we have to acknowledge, is now a lot more so! Keith Devlin, Palo Alto, CA Gary Lorden, Pasadena, CA Contents Introduction The Hero Is a Mathematician? ix 1 Finding the Hot Zone 1 Criminal Geographic Profiling 2 Fighting Crime with Statistics 101 13 3 Data Mining 25 Finding Meaningful Patterns in Masses of Information 4 When Does the Writing First Appear on the Wall? 51 Changepoint Detection 5 Image Enhancement and Reconstruction 63 6 Predicting the Future 77 Bayesian Inference 7 DNA Profiling 89 8 Secrets—Making and Breaking Codes 105 9 How Reliable Is the Evidence? 121 Doubts about Fingerprints 10 Connecting the Dots 137 The Math of Networks viii Contents 11 The Prisoner's Dilemma, Risk Analysis, and Counterterrorism 153 12 Mathematics in the Courtroom 175 13 Crime in the Casino 193 Using Math to Beat the System Appendix Mathematical Synopses of the Episodes in the First Three Seasons of NUMB3RS 207 Index 233 INTRODUCTION The Hero Is a Mathematician ? On January 23, 2005, a new television crime series called NUMB3RS de­ buted. Created by the husband-and-wife team Nick Falacci and Cheryl Heuton, the series was produced by Paramount Network Television and acclaimed Hollywood veterans Ridley and Tony Scott, whose movie credits include Alien, Top Gun, and Gladiator. Throughout its run, NUMB3RS has regularly beat out the competition to be the most watched series in its time slot on Friday nights. What has surprised many is that one of the show's two heroes is a mathematician, and much of the action revolves around mathematics, as professor Charlie Eppes uses his powerful skills to help his older brother, Don, an FBI agent, identify and catch criminals. Many viewers, and several critics, have commented that the stories are entertaining, but the basic premise is far-fetched: You simply can't use math to solve crimes, they say. As this book proves, they are wrong. You can use math to solve crimes, and law enforcement agencies do—not in every instance to be sure, but often enough to make math a powerful weapon in the never-ending fight against crime. In fact, the very first episode of the series was closely based on a real-life case, as we will discuss in the next chapter. Our book sets out to describe, in a nontechnical fashion, some of the major mathematical techniques currently available to the police, CIA, and FBI. Most of these methods have been mentioned during episodes of NUMB3RS, and while we frequently link our explanations to what was depicted on the air, our focus is on the mathematical techniques and how they can be used in law enforcement. In addition we describe X Introduction some real-life cases where mathematics played a role in solving a crime that have not been used in the TV series—at least not directly. In many ways, NUMB3RS is similar to good science fiction, which is based on correct physics or chemistry.