Press MAA Redis Paperback:Layout 1 6/30/11 6:39 PM Page 1 S h a

ematics i Rediscovering Math S i h Shai Simonson m You Do the Mat o n s

llection o R athematics is an eclectic co Rediscovering M n zzles aimed at talented e of mathematical topics and pu lts who want to expand their M d youngsters and inquisitive adu g on problem solving, and i in R view of mathematics. By focus a s to learn the book shows how e couraging rote memorization, t c n h dis d o t stigation, experimentation, h o s h mathematics through inve i n a and teac s excellent text v o s is also an c e thematic m d discovery. Rediscovering Ma i an o e S M m i ll levels. v for training math teachers at a r a e h e i S

e r e of historical periods, with som a h fficulty and cover a wide rang i n Topics range in di n t cs in everyday life, including: t o uncover mathemati g g examples demonstrating how t i o

M c ver the Internet, on to secure communication o D ory and its applicati a s • number the d , an t medieval mathematician Rabbi ial work of a h u • the algebraic and combinator life. e o sports, casinos, and everyday • applications of probability to m Y

atics for a them t cs provides a fresh view of ma Rediscovering Mathemati i ose who c d offers a second chance for th ect, an s those who already like the subj think they don’t. Y o u D o t h e M a t h

ISBN 978-0-88385-770-0

9 780883 857700 MAA i i “master˙color” — 2011/6/30 — 12:23 — page i — #1 i i

Rediscovering Mathematics You Do the Math

i i

i i i i “master” — 2012/5/23 — 11:21 — page ii — #2 i i

c 2011 by the Mathematical Association of America, Inc.

Library of Congress Catalog Card Number 2010939888 Paperback edition ISBN 978-0-88385-780-9 Hardcover edition ISBN 978-0-88385-770-0 Electronic edition ISBN 978-0-88385-912-4 Printed in the United States of America Current Printing (last digit): 10987654321

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page iii — #3 i i

10.1090/clrm/039

Rediscovering Mathematics You Do the Math

Shai Simonson Stonehill College

Published and Distributed by The Mathematical Association of America

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page iv — #4 i i

Committee on Books Frank Farris, Chair Classroom Resource Materials Editorial Board Gerald M. Bryce, Editor Michael Bardzell William C. Bauldry Diane L. Herrmann Barbara E Reynolds Wayne Roberts Susan G. Staples Philip D. Straffin Holly S. Zullo

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page v — #5 i i

CLASSROOM RESOURCE MATERIALS

Classroom Resource Materials is intended to provide supplementary classroom material for students—laboratory exercises, projects, historical information, textbooks with unusual approaches for presenting mathematical ideas, career information, etc.

101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Calculus: An Active Approach with Projects, Stephen Hilbert, Diane Driscoll Schwartz, Stan Seltzer, John Maceli, and Eric Robinson The Calculus Collection: A Resource for AP and Beyond, edited by Caren L. Diefenderfer and Roger B. Nelsen Calculus Mysteries and Thrillers, R. Grant Woods Conjecture and Proof, Mikl´os Laczkovich Counterexamples in Calculus, Sergiy Klymchuk Creative Mathematics, H. S. Wall Environmental Mathematics in the Classroom, edited by B. A. Fusaro and P. C. Kenschaft Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergrad- uate Research, by Hongwei Chen Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E. Weller Geometry From Africa: Mathematical and Educational Explorations, Paulus Gerdes Historical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katz and Karen Dee Michalowicz Identification Numbers and Check Digit Schemes, Joseph Kirtland Interdisciplinary Lively Application Projects, edited by Chris Arney Inverse Problems: Activities for Undergraduates, Charles W. Groetsch Laboratory Experiences in Group Theory, Ellen Maycock Parker Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and Victor Katz Math Made Visual: Creating Images for Understanding Mathematics, Claudi Alsina and Roger B. Nelsen Ordinary Differential Equations: A Brief Eclectic Tour, David A. S´anchez Oval Track and Other Permutation Puzzles, John O. Kiltinen A Primer of Abstract Mathematics, Robert B. Ash Proofs Without Words, Roger B. Nelsen Proofs Without Words II, Roger B. Nelsen Rediscovering Mathematics: You Do the Math, Shai Simonson She Does Math!, edited by Marla Parker Solve This: Math Activities for Students and Clubs, James S. Tanton

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page vi — #6 i i

Student Manual for Mathematics for Business Decisions Part 1: Probability and Simula- tion, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimiza- tion, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Teaching Statistics Using Baseball, Jim Albert Visual Group Theory, Nathan C. Carter Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa Crannell, Gavin LaRose, Thomas Ratliff, Elyn Rykken

MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page vii — #7 i i

Acknowledgments

Thanks to the dozens of teachers and hundreds of students who inspire my passion for do- ing and teaching mathematics. Thanks to all my colleagues who contributed to this effort. Fernando Gouvea contributedto the chapter on How to Read Mathematics. Tim Woodcock and Ralph Bravaco read through many parts of the manuscript, offering excellent sugges- tions, and correcting many of my errors. Bruce Bukiet, Tara Holm, Chris Page, Klaus Peters, and Andrea Simonson reviewed sections of early drafts. Neil Comins, Chet Raymo and Elizabeth Loew shared their experience and wisdom about writing and publishing. Hendrik Lenstra, Art Benjamin, and Ron Rivest provided specific mathematical expertise, and Victor Katz introduced me to the mathematics of Levi ben Gershon. Jerry Bryce and the anonymous CRM review board at the Mathematical Association of America provided many good suggestions and wise guidance. Thanks also to the expert MAA editing and production team of Don Albers, Elaine Pedreira, Carol Baxter, and especially Bev Ruedi. For seven years, the staff at the South Area Solomon Schechter Day School (SASSDS) courageously gave me carte blanche to design and test a curriculum that diverges from common practice. Much of thisbook grew out ofthat experience. My students there blessed me with effort, dedication, respect, and continuing relationships. The administration at Stonehill College granted me a year-long sabbatical to write this book, and the Education department encouraged the use of the book in the design of a new math course for pre-service K–12 teachers, called Mathematical Reasoning for Education (MA 143). The course follows the Massachusetts STEM (science, technology, engineering, and mathematics) initiative, which emphasizes the National Council of Teachers of Math- ematics (NCTM) recommended “student-centered” and “active-learning” approaches. My students in this course create an atmosphere of challenge and excellence, providing me with valuable feedback. Finally, thanks to wife Andrea, and my children Zosh, Yair, and Yona who humor my passions and feed me with love.

vii

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page viii — #8 i i

To my wife and companion Andrea, always willingto share my joy in solving puzzles large and small, who taught me that love is a puzzle best enjoyed unsolved.

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page ix — #9 i i

Contents

Acknowledgements vii A Guide for the Reader xiii Introduction: How to Read Mathematics xvii 1 An Example of Mathematical Writing—TheBirthday Paradox ...... xxii 2 OurReaderTacklestheBirthdayParadox ...... xxiii 3Challenges ...... xxix 1 MathematicalDiscoveryintheClassroom 1 1.1 ASimpleLesson...... 1 1.2 TheMathTeacherasConductor...... 2 1.3 TheEffectiveTeacher ...... 9 1.4 Challenges...... 10 2 Don’t Reach for Your Calculator (Yet) 13 2.1 AMagicTrick...... 18 2.2 CleverCalculations ...... 20 2.3 PythagoreanTriples ...... 22 2.4 Challenges...... 26 3 Have Another Piece of Pie, Zeno? 31 3.1 ASimplerWay ...... 34 3.2 Euclid,Proofs,andWritingMathematics ...... 35 3.3 Challenges...... 38 4 Thinking Like a Mathematician—Lessons from a Medieval Rabbi 41 4.1 RabbiLevibenGershonandhisSums ...... 44 4.2 TheNumberofSquaresandRectanglesinaGrid ...... 52 4.3 The Triangle Puzzle—A Model Mathematical Problem ...... 55 4.4 Challenges...... 61 5 What is Mathematics Good For? 63 5.1 KnockHockey...... 64 5.2 Basketball ...... 66

ix

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page x — #10 i i

x Contents

5.3 RatesofGrowth...... 69 5.4 Challenges...... 72 6 Three Averages 77 6.1 AverageI.TheArithmeticAverage ...... 77 6.2 AverageII.TheHarmonicAverage ...... 78 6.3 AverageIII.TheGeometricAverage ...... 79 6.4 ExploringAverages ...... 82 6.5 Challenges...... 89 7 Algorithms—The Unexpected Role of Pure Mathematics 93 7.1 ASolutiontotheTwoJugPuzzle ...... 94 7.2 Euclid’sAlgorithm ...... 96 7.3 Other Methods for Calculating Greatest Common Divisors ...... 99 7.4 TheEfficiency(orSpeed)ofanAlgorithm ...... 99 7.5 TheEgyptianMultiplicationAlgorithm...... 101 7.6 TheFastModularExponentiationAlgorithm ...... 103 7.7 Greatest Common Divisors, Algorithms, and E-Commerce ...... 105 7.8 E-CommerceandCryptography...... 106 7.9 TheRestoftheStory—TheRSAAlgorithm ...... 108 7.10Challenges...... 109 8 Pythagoras’ Theorem and Math by Pictures 113 8.1 AProofofPythagoras’Theorem ...... 113 8.2 APersonalExperience ...... 115 8.3 TheHeightofaPyramid ...... 120 8.4 Challenges...... 124 9 Memorizing Versus Understanding 129 9.1 Example1:FOIL ...... 129 9.2 Example2:SquareRoots ...... 131 9.3 Example3:TheQuadraticEquation ...... 133 9.4 A“RealLife”QuadraticEquation ...... 138 9.5 Challenges...... 140 10 Games and Gambling 143 10.1ACarnivalGame ...... 143 10.2Craps...... 146 10.3TheCarelessCasino...... 151 10.4Challenges...... 152 11 Soccer Balls and Counting Tricks 159 11.1Euler’sFormula ...... 160 11.2PlatonicSolids...... 162 11.3Counting...... 163 11.4ThePuzzleoftheSoccerBall ...... 164 11.5Challenges...... 166 12 Pizza Pi and Area 169 12.1PiintheBible ...... 170

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xi — #11 i i

Contents xi

12.2TheAreaofaCircle...... 173 12.3AHardPuzzleInvolvingAreas ...... 175 12.4Challenges...... 179 13 Back to the Classroom 185 13.1AClassroomExperience ...... 186 13.2EmphasizingUnderstanding...... 188 13.3WhatNext? ...... 193 Resources for Rediscovering Mathematics 195 Further Reading 199 Index 201 About the Author 207

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xii — #12 i i

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xiii — #13 i i

A Guide for the Reader

What does a mathematician do? Most people know that scientists design experiments to verify theories of how the world works; writers write to communicate, educate, or evoke emotions and thoughts; politicians influence policy to hopefully make our lives better; and musicians perform or compose to bring joy and meaning to their lives and to others. But mathematicians? Few people have any notion of what a mathematician does day to day. Many people associate mathematics with arithmetic, terminology, memorization, drills and methods, but there is much more to mathematics. If youasked a mathematician, youmight be surprised to hear her say that what she does for a livingfeels closer to writing poetry than doing rote calculations. Indeed, mathematics is more about planning,exploring, creating, and experimenting than it is about memorizing and labeling. It is elegant, logical, beautiful and inspiring. Mathematicians solve problems; they search for structure and truth; they seek to understand why shapes, ideas, numbers and patterns interact and behave the way they do.

What is this Book About? The underlying theme of this book is that mathematics education should focus less on rote memorization of terminology and algorithms, and more on understanding and deep com- prehension that comes through investigation, experiment, and personal exploration. This book is about rediscovering mathematics. Most of you have already discovered math—the formulas and terminology learned in school—but few have had the joy of the wider world of mathematics, filled with creativity, exploration, experiment, and variety. Indeed, I hope this book allows you to rediscover mathematics with the maturity, perspective and patience that come with distancing yourself from your past experiences. The book does not survey all of mathematics, nordoes itconcentrate on just one special area. It is a flight over the landscape of mathematics with occasional stops at places that look interesting. Topics include well-known puzzles, obscure ones, and a few published here for the first time. Each puzzle is focused on a particular theme, intended to challenge the reader to discover its secrets. Challenges are presented as the secrets are slowly discov- ered and the puzzles are unraveled. You are given the opportunityto try to figure out the next part of a puzzle yourself, or just read the solution and continue. The challenges are

xiii

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xiv — #14 i i

xiv A Guide for the Reader

meant to entertain, educate, and motivate. Along the way, there is advice on how to study, read and teach math. Both specific mathematical topics and problem-solvingtechniques are presented. With every challenge comes a blueprint for how an experienced mathematician might have solved it. For further practice and exploration, there are additional challenges at the end of each chapter, without the solutions provided.

Is this Book for You? Are youa teacher? This bookwillhelpyou make mathematics come alive for your students. Your students will understand more, memorize less, and enjoy the process of discovering mathematics. Depending on the level of your students, some lesson plans can come directly from your reading, but even when the specific mathematics is too advanced for your stu- dents, you can design your own dynamic lessons usingthe examples in thebook as models. The effort you expend in reading this book and rediscovering mathematics will empower you to pass on your knowledge and reshape the popular perception of mathematics —one child at a time. The book is not just for teachers or prospective teachers. It is for anyone who is intel- lectually curious and looking for a guide to help revisit and reconsider mathematics. It is for the math whiz looking for interesting topics outside the standard fare, as well as for the math phobic willing to give mathematics another chance.

Preliminaries Rediscovering mathematics is not a spectator sport; you must get involved; you must be an active participant. And like any sport, in order to participate, you must first get in shape. Mathematical fundamentals need to be mastered before any discoveries can be accom- plished. Getting in shape mathematically may be hard work, but the reward is personal mathematical discovery that brings joy and builds confidence. When you first take golf lessons, you spend a lot of time training your body to move in the right way. Impatience causes some people to just go out and hit the ball, but they develop bad habits that are hard to unlearn. Golf is a complex sport with technique and strategy, and the way to best appreciate it is to study the fundamentals, and then play, and play some more. Mathematics is the same as golf: master the fundamentals and then practice their application on various challenges. For this book, the fundamentals you are expected to know include: arithmetic of whole numbers and fractions, a little terminology, basic algebra, and elementary geometry—the same set of topics expected of you for the SAT. The challenges that appear throughout the book are your opportunity to practice. Try every challenge, but be patient. Not every challenge will yield its secret easily. The challenges vary in difficulty from the routine to the subtle. Even if you solve just a few of the challenges, your failed attempts will not be in vain. It took Andrew Wiles eight years to complete his proof of Fermat’s Last Theorem, a problem that had baffled mathematicians for centuries. Indeed, after seven years, Wiles presented a “proof” at an international conference only to later find a flaw. Wiles, along with Richard Taylor, took another year to untangle the error. You won’t need eight years to

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xv — #15 i i

A Guide for the Reader xv

solve any challenge presented here, but there will be times when you get stuck. Often, you may not be able to discern how a particular technique or skill helps you solve a problem. Have faith in yourself and in the subject. “Wax on, wax off,” Mr. Miagi tells his impatient young student in the movie The Karate Kid. The boy complains that he wants to start learning karate and not just wax cars. Later he learns that the motion he has used over and over to wax the cars has taught him the instinctive motion to block an attack. For Andrew Wiles, it was precisely an earlier failed effort that showed him the way topatch up the flaw in his proof of Fermat’s Last Theorem.1 He summarized his long battle with Fermat’s Last Theorem in this way: “However impenetrable it seems, if you don’t try it, then you can never do it.” Patience.

Why Bother? Math can be hard, the fundamentals challenging, and other pursuits are easier and more fun. So why bother? The answer is because mathematics is an intrinsically beautiful sub- ject. The study and practice of mathematics can raise your spirits, gladden your heart, and put a smile on your face. However, even if your appreciation of the subject never rises to a level of passion, be assured that studying mathematics makes you sharper. Exploring math- ematical challenges is the intellectual equivalent of physical conditioning. After working through this book, you will analyze more quickly, think more critically, and ponder things you never used to even notice. You will become someone not easily fooled.

How to Use this Book Try hard not to read the solutions to the challenges until you have explored, investigated, and tried to solve them on your own. In a classroom setting, I have students work in groups on a challenge, and follow up by discussing alternative approaches and evaluating each group’s progress and effectiveness. We review “solutions” only after each group has sub- mitted a journal of their attempts, successes, failures, and discoveries. There are many ways to attack each challenge in this book, and the “solutions”provided represent just one option.You may find that you have a more elegant insight to a challenge or a better solution. Trust yourself—you probably do. When I teach using this book, the class often discovers new and interesting approaches that I had not considered. Finally, solutionsto challenges almost always involvewriting,rather than a simple “answer” likean equation or a number. You must be convincing, logical, organized, and rigorous. Learning to write effectively and clearly is as important in mathematics as it is in every academic endeavor. See the discussion in Chapter 3 about proofs and writing mathematics. This is the sort of book that should not be read in one sitting. Keep it on your night table with a pad of paper nearby, and work throughit at your leisure. I recommend that you start by reading the Introduction about how to read mathematics. After that, feel free to skip around from chapter to chapter. The chapters are mostly independent, and sticking to what you like is more important than any linear overarching structure. The only exception is that Chapter 10 depends slightly on the material in Chapter 3.

1NOVA Online: The Proof, WGBH, 1997.

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xvi — #16 i i

xvi A Guide for the Reader

All the chapters mix technical material with pedagogical advice, but Chapters 1, 5, 9, 12, and 13 have a more pedagogical focus than the other chapters. Chapters 1 and 13 respectively introduce and revisit the main themes of the book. Chapters 4 and 7 are the most mathematically difficult chapters, and are appropriate for more ambitious readers. If you are looking for particular mathematical topics, the following list will help you choose a chapter.

Probability IntroductionandChapter10 Algebra Chapters2,3,4,5,6,7,9,10,and11 Geometry Chapters5,6,8,9,and12 Number Theory Chapters 1,2, 3,4, and 7 Algorithms Chapters1,2,3,and7 Combinatorics Chapters 4, 10, and 11 The book can be picked up again and again, each time providing you with a new ex- perience. If you get frustrated or tired, put the book down and get some ice cream. Come back again when you are happy and ready for a new challenge.

— SS Sharon, MA

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xvii — #17 i i

Introduction: How to Read Mathematics

A reading protocol is a set of strategies that a reader must use in order to benefit fully from reading the text. Poetry calls for a different set of strategies than fiction, and fiction a different set than non-fiction. It would be ridiculous to read fiction and ask oneself what is the author’s source for the assertion that the hero is blond and tanned; it would be wrong to read non-fiction and not ask such a question. This reading protocol extends to a viewing or listening protocol in art and music. Indeed, much of the introductory course material in literature, music, and art is spent teaching these protocols. Ed Rothstein’s book, Emblems of Mind, a fascinating book that focuses on the re- lationship between mathematics and music, touches implicitly on reading protocols for mathematics. [Mathematics is] “a language that can neither be read nor understood without initia- tion.”2 Mathematics has a reading protocol all its own, and just as we learn how to read a novel or a poem, listen to music, or view a painting, we should learn to read mathematics. When we read a novel we become absorbed in the plot and characters. We try to follow the various plot lines and how each affects the development of the characters. We make sure that the characters become real people to us, both those we admire and those we despise. We do not stop at every word, but imagine the words as brushstrokes in a painting. Even if we are not familiar with a particular word, we can still see the whole picture. We rarely stop to think about individual phrases and sentences. Instead, we let the novel sweep us along with its flow, and carry us swiftly to the end. The experience is rewarding, relaxing, and thought provoking. Novelists frequently describe characters by involving them in well-chosen anecdotes, rather than by describing them by well-chosen adjectives. They portray one aspect, then another, then the first again in a new light and so on, as the whole picture grows and comes more and more into focus. This is the way to communicate complex thoughts that defy precise definition. Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space. Both a mathematics article and a novel are tellinga story

2Emblems of Mind, Edward Rothstein, Harper Perennial, 1996, p. 15.

xvii

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xviii — #18 i i

xviii How to Read Mathematics

and developing complex ideas, but a math article does the job with a tiny fraction of the words and symbols of those used in a novel. The beauty in a novel is in the aesthetic way it uses language to evoke emotions and present themes which defy precise definition. The beauty in a mathematics article is in the elegant way it concisely describes precise ideas of great complexity. What are the common mistakes people make in trying to read mathematics, and how can these mistakes be corrected?

Don’t Miss the Big Picture “Reading Mathematics is not at all a linear experience ...Understanding the text re- quires cross references, scanning, pausing and revisiting.”3

Don’t assume that understanding each phrase will enable you to understand the whole idea. This is like trying to see a portrait by staring at each square inch of it from the distance of your nose. You will see the detail, texture and color but miss the portrait completely. A math article tells a story. Try to see what the story is before you delve into the details. You can go in for a closer look once you have built a framework of understanding. Do this just as you might reread a novel.

Don’t be a Passive Reader “A three-line proof of a subtle theorem is the distillation of years of activity. Reading mathematics ...involves a return to the thinkingthat went into the writing.”4

Explore examples for patterns. Try special cases. A math article usually tells only a small piece of a much larger and longer story. The author usually spends months exploring things and going down blind alleys. After a period of exploration, experiment, and discovery, the author organizes his/her conclusions into a story that covers up all the mistakes, wrong turns, and associated motivation, presenting the completed idea in a neat linear flow. The way to deeply understand the author’s idea is to recreate what the author left out. There is a lot between the lines of a polished mathematical exposition. The reader must participate. At every stage, he/she must decide whether or not the idea being presented is clear. Ask yourself these questions: Why is this idea true?  Do I really believe it?  Could I convince someone else that it is true?  Why didn’t the author use a different argument?  Do I have a better argument or method of explaining the idea?  Why didn’t the author explain it the way that I understand it?  Is my way wrong?  Do I really get the idea?  3Emblems of Mind, p. 16. 4ibid, p. 38.

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xix — #19 i i

How to Read Mathematics xix

Am I missing some subtlety?  Did the author miss a subtlety?  If I can’t understand the idea, can I understand a similar but simpler concept?  If so, which simpler concept?  Is it really necessary to understand this idea?  Can I accept this point without understanding the details of why it is true?  Will my understandingof the whole storysuffer from not understandingwhy the point  is true? Puttingtoo littleeffort intothis participationis like reading a novel without concentrat- ing. After half an hour, you realize the pages have turned, but you have been daydreaming and don’t remember a thing you read.

Don’t Read Too Fast Reading mathematics too quickly results in frustration. When reading a novel, a half hour of concentration might net the average reader 20–60 pages with full comprehension, de- pending on the novel and the experience of the reader. The same half hour buys you 0–10 lines of a math article, depending on the article and how experienced you are at reading mathematics. For example, consider the following theorem from Levi ben Gershon’s book, Maaseh Hoshev (The Art of Calculation),written in 1321. “When you add consecutive numbers starting with one, and the number of numbers you add is odd, the result is equal to the product of the middle number among them times the last number.” It is natural for modern day mathematicians to write this as

2k 1 C i .k 1/.2k 1/: D C C i 1 XD A reader should take as much time to unravel the two-inch version as he would to unravel the two-line version.

Challenge What does the expression 2k 1 C i i 1 XD mean?

Solution Given an integer k, addup thevalues of i, as i ranges from 1 through 2k 1. C In other words, 1 2 3 .2k 1/.  C C C


C C Challenge Explain the meaning of 2k 1 in the expression C 2k 1 C i‹ i 1 XD

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xx — #20 i i

xx How to Read Mathematics

Solution It represents an odd number. Every even number is two times something, so every odd number is two times something plus one. 

Challenge On the right side of the equation

2k 1 C i .k 1/.2k 1/; D C C i 1 XD the term 2k 1 represents the last number in the sum. What does k 1 represent and why? C C Solution The expression k 1 represents the middle number in the sum. The middle C number is halfway between 1 and 2k 1. That is, the middle number equals C .1 2k 1/=2 .2k 2/=2 k 1:  C C D C D C Challenge Can you provide an illustration of Levi’s theorem?

Solution An illustration of Levi’s theorem is

2.2/ 1 C i 1 2 3 4 5 3 5: i 1 D D C C C C D  In this case, k 2.  X D Challenge Why is this theorem true?

Solution The following proof is from one of my students. Her idea is to pair up all the numbers except the last, creating k pairs each of which sums to 2k 1. Start with the C middle pair, k and k 1. This pair sums to 2k 1. Continue pairing numbers moving left C C from k and right from k 1. Each new pair also sums to 2k 1 since moving left subtracts C C one and moving right adds one. The last pair is 1 and 2k, giving k pairs all together. The last number, 2k 1, is left unpaired. The total sum equals the sum of the k pairs plus the C last number, 2k 1. That is, the total sum equals C k.2k 1/ 2k 1 .k 1/.2k 1/:  C C C D C C In contrast, here is Levi’s elegant proof discussed in Chapter 4. Levi’s proof is similar to my student’s but he pairs up numbers startingwith the pair surrounding the middle term, k and k 2, and working outward. He points out that each pair sums to twice the middle C term. This continues until the final pair of numbers, 1 and 2k 1. Therefore, the entire C sum is the same as if every one of the 2k 1 terms were the middle term, k 1. That is, C C the sum is .k 1/.2k 1/. C C You can speed up your math reading skill by practicing, but there is no shortcut. Like learning any skill,trying too much too fast can set you back, and may kill your motivation. Imagine joininga high-energy aerobics class when you have not worked out for two years. You may make it throughthe first class, but you are not likelyto come back. The frustration from seeing the experienced class members effortlessly do twice as much as you,whileyou moan the wholenext day from soreness, might be too much to take. Be realistic, be patient, and don’t punish yourself.

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xxi — #21 i i

How to Read Mathematics xxi

Make the Idea your Own The best way to understand what you are reading is to make the idea your own. This means following the idea back to its origin, and rediscovering it for yourself. Mathematicians often say that to understand something you must first read it, then write it down in your own words, then teach it to someone else. Everyone has a different set of tools and a different level of “chunking up” complicated ideas. Make the idea fit in with your own perspective and experience.

‘‘When I use a word, it means just what I choose it to mean"5 “The meaning is rarely completely transparent, because every symbol or word already represents an extraordinary condensation of concept and reference”6

A well-written mathematical text will be careful to use a word in one sense only, mak- ing a distinction, say, between combination and permutation (or arrangement ). A strict mathematical definition might imply that “yellow rabid dog” and “rabid yellow dog” are different arrangements of words but the same combination of words. Most English speak- ers would disagree. This extreme precision is utterly foreign to most fiction and poetry writing, where using multiple words, synonyms, and varying descriptions is de rigueur.A reader is expected to know that an absolute value is not about some value that happens to be absolute, nor is a function about anything functional. A particular notorious example of a phrase commonly used in mathematical writing that might easily be misinterpreted is the use of “It follows easily that” and equivalent constructs. The phrase means something like this: One can now check that the next statement is true with a certain amount of essen- tially mechanical, though perhaps laborious, checking. I, the author, could do it, but it would use up a large amount of space and perhaps not accomplish much, since it’d be best for you to go ahead and do the computation to clarify for yourself what’s going on here. I promise that no new ideas are involved, though of course you might need to think a littlein order to find just the right combination of good ideas to apply. In other words, the construct “It follows easily that,” when used correctly, is a signal to the reader that what’s involved here is perhaps tedious and even difficult, but involves no deep insights. The reader is then free to decide whether the level of understanding desired requires going through the details or instead, warrants saying “Okay, I’ll accept your word for it.” Now, regardless of your opinion about whether that construct should be used in a par- ticular situation, or whether authors always use it correctly, you should understand what it is supposed to mean. “It follows easily that” does not mean if you can’t see this at once, you’re a dope, nor does it mean this shouldn’t take more than two minutes,

5Humpty Dumpty to Alice in Through the Looking Glass by Lewis Carroll. 6Emblems of Mind, p. 16.

i i

i i i i “master” — 2012/5/23 — 10:51 — page xxii — #22 i i

xxii How to Read Mathematics

but a person who doesn’t know the lingo might misinterpret the phrase, and thereby feel discouraged. This is apart from the issue that one person’s tedious task is another person’s challenge, so not only must the audience engage the author, but the author must correctly judge the audience.

Know Thyself Texts are written with a specific audience in mind. Make sure that you are the intended audience, or be willing to do what it takes to become the intended audience. For example, take T.S. Eliot’s A Song for Simeon: Lord, the Roman hyacinths are blooming in bowls and The winter sun creeps by the snow hills; The stubborn season has made stand. My life is light, waiting for the death wind, Like a feather on the back of my hand. Dust in sunlight and memory in corners Wait for the wind that chills towards the dead land. Eliot’s poem pretty much assumes that a reader is going to either know who Simeon was or be willing to find out. It also assumes a reader will be somewhat experienced in reading poetry and/or is willing to work to gain such experience. Eliot assumes that a reader will either know or investigate the allusions here. This goes beyond knowledge of things like who Simeon was. For example, why are the hyacinths “Roman?” Why is that important? Eliot assumes that the reader will read slowly and pay attention to the images: he juxta- poses dust and memory, relates old age to winter, compares waitingfor death with a feather on the back of the hand, and so on. He assumes that a reader will recognize this as poetry; in a way, Eliot is assuming that the reader is familiar with a whole poetic tradition. For example, a reader is supposed to notice that alternate lines rhyme, but that the others do not. Most of all, the poet assumes that a reader will read not only with the mind, but also with his/her emotions and imagination, allowing the images to summon up this old man, tired of life but hanging on, waiting expectantly for some crucial event, for something to happen. Most math books are written with the assumption that the audience knows certain things, that they have a certain level of “mathematical maturity,” and so on. Before you start to read, make sure you know what the author expects you to know.

1 An Example of Mathematical Writing—The Birthday Paradox To allow an opportunity to experiment with the guidelines presented here, I am including a small piece of mathematics often called the birthday paradox. The first part is a concise mathematical article explaining the problem and solving it. The second is an imaginary Reader’s attempt to understand the article by using the appropriate reading protocol. This article’s topic is probability and is accessible to a bright and motivated reader.

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xxiii — #23 i i

How to Read Mathematics xxiii

A professor offers to bet anyone in a class of 30 random students that there are at least two people in the class with the same birthday (month and day, but not necessarily year). Would you accept the bet? What if there were fewer people in the class? Let the birthdays of n people be uniformly distributed among 365 days of the year (for simplicity assume no leap years ). We prove that the probability that at least two people have the same birthday (month and day) equals

365 364 363 .365 n 1/ 1   


 C : 365n  à What is the probabilitythat among 30 students in a room, there are at least two or more with the same birthday? For n 30, the probability of at least one matching birthday is D about 71%. This means that with 30 people in your class, the professor should win the bet 71 times out of 100 in the long run. It turns out that with 23 people, she should win about 50% of the time. Here is the proof: Let P.n/ be the probability in question. Let Q.n/ 1 P.n/ D be the probability that no two people have a common birthday. Now calculate Q.n/ by dividingthe number of n birthdayswithoutany duplicatesby the total number of n possible birthdays. Then solve for P.n/. The number of n birthdays without duplicates is

365 364 363 .365 n 1/:   


 C This is because there are 365 choices for the first birthday, 364 for the next and so on for n birthdays. The total number of n birthdays without any restriction is 365n because there are 365 choices for each of n birthdays. Therefore, Q.n/ equals

365 364 363 .365 n 1/   


 C : 365n Solving for P.n/ gives P.n/ 1 Q.n/ and hence our result. D 2 Our Reader Tackles the Birthday Paradox In this section, a naive Reader tries to make sense out of the last few paragraphs. My Reader may seem to catch on to things relatively quickly. However, be assured that in reality a great deal of time passes between each of my Reader’s comments, and that I have left out many of the Reader’s remarks that explore dead-end ideas. To experience what the Reader experiences requires much more than just reading through his/her lines. Think of his/her part as an outline for your own efforts.

Know Thyself

Reader (R) I don’t know anything about probability, can I still make it through? Professional (P) Let’s give it a try. We may have to backtrack a lot at each step. R What does the phrase “30 random students” mean?

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xxiv — #24 i i

xxiv How to Read Mathematics

‘‘When I use a word, it means just what I choose it to mean" P Good question. It doesn’t mean that we have 30 spacey or scatter-brained students. It means we should assume that the birthdays of these 30 people are independent of one another and that every birthday is equally likely for each person. The author writes this more technically a little further on: “Let the birthdays of n people be uniformly distributed among 365 days of the year.” R Isn’t that obvious? Why bother saying that? P Yes, the assumption is fairly obvious. The author is just setting the groundwork. The sentence guarantees that every birthday is equally likely and the solution does not involve some imaginative fanciful science-fiction. R What do you mean? P For example, the author is not looking for a solution like this: the 30 people are at a convention for identical twins, so the probability of two or more people with the same birthday is 100%. Though this solution may be clever, it sidesteps the intended mathe- matical question. Incidentally, the assumption “let the birthdays of n people be uniformly distributed among 365 days of the year” also implies that nobody in this problem is born on February 29. A date that comes once every four years would be less common than other birthdates, so for simplicity we exclude it. R I don’t understand that long formula, what’s n? P The authoris solvingthe problem for any number of people, notjustfor 30. The author, from now on, is going to call the number of people n. R I still don’t get it. So what’s the answer?

Don’t Be a Passive Reader—Try Some Examples P Well, if you want the answer for 30, just set n 30. D R Okay, but that looks complicated to compute. Where’s my calculator? Let’s see: 365  364 363 336. That’s tedious, and the final exact value won’t even fit on my  


 calculator. It reads

2:1710301835085570660575334772481e 76: C If I can’t even calculate the answer once I know the formula, how can I possibly understand where the formula comes from? P You are right that this answer is inexact, but if you actually go on and do the division, your answer won’t be too far off. R The whole thing makes me uncomfortable. I would prefer to be able to calculate it more exactly. Is there another way to do the calculation? P How many terms in your product? How many terms in the product on the bottom? R You mean 365 is the first term and 364 is the second? Then there are 30 terms. There are also 30 terms on the bottom, (30 copies of 365). P Can you calculate the answer now?

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xxv — #25 i i

How to Read Mathematics xxv

R Oh, I see. I can pair up each top term with each bottom term, and do 365=365 as the first term, then multiply by 364=365, and so on for 30 terms. This way the product never gets too big for my calculator. (After a few minutes) ...Okay, I got 0:29368, rounded to 5 places. P What does this number mean?

Don’t Miss the Big Picture R I forgot what I was doing. Let’s see. I was calculating the answer for n 30. The D 0:29368 is everything except for subtracting from 1. Subtracting from 1, I get 0:70632. Now what does that mean? P Knowing more about probabilitywould help, but this simply means that the probability that two or more out of the 30 people have the same birthday is 70;632 out of 100;000 or about 71%. R That’s interesting. I wouldn’t have guessed that. You mean that in my class with 30 students, there’s a pretty good chance that at least two students have the same birthday? P Yes that’s right. You might want to take bets before you ask everyone their birthday. Many people don’t expect a duplicate to occur. That’s why some authors call this the birth- day paradox. R So that’s why I should read mathematics, to make a few extra bucks? P I see how that might give you some incentive, but I hope the mathematics also inspires you without the monetary prospects. R I wonder what the answer is for other values of n. I will try some more calculations. P That’s a good idea. We can even make a picture out of all your calculations. We could plota graph of the number of people versus the probabilitythat a duplicate birthday occurs, but maybe this can be left for another time. R Oh look, the author did some calculations for me. He says that for n 30 the answer D is about 71%; that’s what I calculated too. And, for n 23 it’s about 50%. Does that make D sense? I guess it does. The fewer people there are, the smaller the chance of a common birthday. Hey, I am anticipating the author. Pretty good. Okay, let’s go on. P Good, now you’re telling me when to continue.

Don’t Read Too Fast R It seems that we are up to theproof.This must explain why that formula works. What’s this Q.n/? I guess that P stands for probability but what does Q stand for? P The author is defining something new. He is using Q just because it’s the next letter after P , but Q.n/ is also a probability, and closely related to P.n/. It’s time to take a minute to think. What is Q.n/ and why is it equal to 1 P.n/? R Q.n/ is the probabilitythat no two people have the same birthday. Why does the author care about that? Don’t we want the probability that at least two have the same birthday? P Good point. The author doesn’t tell you this explicitly, but between the lines, you can infer that he has no clue how to calculate P.n/ directly. Instead, he introduces Q.n/ which supposedly equals 1 P.n/. Presumably, the author will proceed next to tell us how to

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xxvi — #26 i i

xxvi How to Read Mathematics

compute Q.n/. By the way, when you finish this article, you may want to deal with the problem of calculating P.n/ directly. That’s a perfect follow up to the ideas presented here. R First things first. P Ok. So once we know Q.n/, then what? R Then we can get P.n/. Because if Q.n/ 1 P.n/, then P.n/ 1 Q.n/. Fine, D D but why does Q.n/ equal 1 P.n/? Does the author assume this is obvious? P Yes, he does, but what’s worse, he doesn’t even tell us that it is obvious. Here’s a rule of thumb: when an author says clearly this is true or this is obvious, if you want to make sure you understand why it is obvious, take 15 minutes to convince yourself. If an author doesn’t explicitly say that something is obvious, but indirectly suggests that it is, take a little longer. R How do I know when I should stop and think? P Just be honest with yourself. When in doubt, stop and think. When too tired, go watch television. R So why does Q.n/ equal 1 P.n/? P Let’s consider a special case. If the probability of getting two or more of the same birthdays is 1=3, then what’s the probabilityof not getting two or more? R It’s 2=3, because the probability of something not happening is the opposite of the probability of it happening.

Make the Idea Your Own P Well, you should be careful when you say things like opposite, but you are right. In fact, you have discovered one of the first rules taught in a course on probability. Namely, that the probability that something will not occur is 1 minus the probability that it will occur. Now you are ready to continue to the next paragraph in the article. R It seems to be explaining why Q.n/ is equal to the long complex-looking formula shown. I will never understand this. P The formula for Q.n/ is tough to understand and the author is counting on your dili- gence, persistence, and/or background here to get you through. R He seems to be counting all possibilities of something and dividing by the total possi- bilities, whatever that means. I have no idea why. P Maybe I can fill you in here on some background before you investigate any further. Given a number of equally likely outcomes, the probabilityof the occurrence of a particular event is defined inmathematics to be: the totalnumber of outcomes that constitutethe event divided by the total number of possible outcomes. For example, the probability that you throw a four when throwing a die is 1=6, because there is one possible four, and there are six equally likely possible outcomes. What’s the probabilitywhen throwing a die, that you throw a four or a three? R Well I guess 2=6 (or 1=3) because the total number of outcomes is still six but I have two possible outcomes that work.

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xxvii — #27 i i

How to Read Mathematics xxvii

P Good. Here’s a harder example. What about the probability of throwing a sum of four when you roll two dice? There are three ways to get a four (1–3, 2–2, 3–1) while the total number of possible outcomes is 36. That is 3=36 or 1=12. Look at the following 6 by 6 table and convince yourself. 1–1, 1–2, 1–3, 1–4, 1–5, 1–6 2–1, 2–2, 2–3, 2–4, 2–5, 2–6 3–1, 3–2, 3–3, 3–4, 3–5, 3–6 4–1, 4–2, 4–3, 4–4, 4–5, 4–6 5–1, 5–2, 5–3, 5–4, 5–5, 5–6 6–1, 6–2, 6–3, 6–4, 6–5, 6–6 What about the probability of throwing a sum of seven? R Wait. What does 1–1 mean? Doesn’t that equal 0? P Sorry, my bad. I was using the minus sign as a dash, just to mean a pair of numbers, so 1–1 means a roll of one on each die—snake eyes. R Couldn’t you have come up with a better notation? P Well maybe I could/should have, but commas would look worse, a slash would look like division, and anything else might be just as confusing. We aren’t going to publish this transcript anyway. R That’s a relief. Well, I know what you mean now. To answer your question, I can get a seven in six ways: 1–6, 2–5, 3–4, 4–3, 5–2, or 6–1. The total number of outcomes is still 36, so I get 6=36 or 1=6. That’s interesting. The chance of rolling a sum of seven is higher than the chance of rollinga sum of four. The sums are not equally likely. P Excellent. Indeed, when rolling two dice, a sum of seven occurs more frequently than rolling a sum of four. The situation would be very different if we were spinning a wheel with the numbers 2 through 12 written in equally spaced intervals. In that case, each one of the eleven values would have probability 1=11; and a seven would occur exactly as often as a four. R Okay, now I am an expert. Is probability just about counting? P Sometimes, yes. But counting things is not always so easy. R I see, let’s go on. By the way, did the author really expect me to know all this? My friend took a course called Probability and Statistics; I bet he knows all this stuff. P There’s a lot of information implied in a small bit of mathematics. Yes, the author ex- pected you to know all this, or to discover it yourself just as we have done. If I hadn’t been here, you would have had to ask yourself these questions, and answer them by thinking, looking in a reference book, or consulting your friend. R So the probability that there are no two people with the same birthday is the number of possible sets of n birthdays without a duplicate divided by the total number of possible sets of n birthdays. P Excellent summary. R I don’t like using n, so let me use 30. Perhaps, then, the generalization to n will be easy to see.

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xxviii — #28 i i

xxviii How to Read Mathematics

P Great idea. It is often helpful to look at a special case before tackling the general case. R So how many sets of 30 birthdaysare there total? I can’t do it. I guess I need to restrict my view even more. Let’s pretend there are only two people. P Fine. Now you’re thinking like a mathematician. Let’s try n 2. How many sets of D two birthdays are there total? R I number the birthdays from 1 to 365 and forget about leap years. These are all the possibilities:

1–1, 1–2, 1–3, ..., 1–365, 2–1, 2–2, 2–3, ..., 2–365, : : : : : : : :


365–1, 365–2, 365–3, ..., 365–365. P When you write 1–1, do you mean 1 1 0, as in subtraction? D R Very funny. You know exactly what I mean. P Yes I do, and nice choice of notation I might add. Now how many pairs of birthdays are there? R There are 365 365 total possibilities for two people.  P And how many are there when there are no duplicate birthdays? R I can’t use 1–1, or 2–2, or 3–3 or ...365–365, so I get

1–2, 1–3, ..., 1–365, 2–1, 2–3, ..., 2–365, : : : : : :


365–1, 365–2, ..., 365–364 The total number here is 365 364 since each row now has 364 pairs instead of 365.  P Good. You are going a little quickly here, but you’re 100% right. Can you generalize now to 30? What is the total number of possible sets of 30 birthdays? Take a guess. You’re getting good at this. R Well if I had to guess, (it’snot really a guess, after all, I already know the formula), I would say that for 30 people you get 365 365 365, 30 times, for the total number  


 of possible sets of birthdays. P Exactly. Mathematicians write 36530. And what is the number of possible sets of 30 birthdays without any duplicates? R I know the answer should be 365 364 363 362 336, (that is, start at 365 and    


 multiplyby one less for 30 times), but I am not sure I really see why this is true. Perhaps I should do the case with three people first, and work my way up to 30? P Splendid idea. Let’s quit for today. The whole picture is there for you. When you are rested and you have more time, you can come back and fill in that last bit of understanding. R Thanks a lot; it’s been an experience. Later.

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xxix — #29 i i

How to Read Mathematics xxix

What Now? Now you are ready to read mathematics. You also learned some fundamentals of probabil- ity. Before you jump ahead to Chapter 1, you can test what you learned by trying some of the challenges that follow. Or if you prefer, come back to these challenges after completing Chapter 10, which contains a more in-depth discussion of probability. Chapter 1, which follows next, sets the theme of rediscovering mathematics. It de- scribes a classroom in which mathematics is learned through exploration and investigation, rather than through memorization and rote.

3 Challenges 1. What is the smallest number of people for which the probability that at least two of them have the same birthday is: a. 100%? b. at least 90%? c. at least 10%? 2. What is the probabilitythat two randomly chosen people were born on different days of the week? What is the probabilitythat they were born on the same day? 3. What is the probabilitythat three randomlychosen people were born on different days of the week? What is the probability that at least two of the three were born on the same day of the week? 4. How many randomly chosen people are needed in order to guarantee that there is a probabilityof at least 80% that two or more of the people were born on the same day of the week? How many people are needed to guarantee at least a 50% probability of two or more being born on the same day of the week? 5. When you roll two dice, what is the probabilitythat (a) the two dice sum to five? (b) at least one of the dice is a five? (c) at least one of the dice is an even number? (d) the sum of the two dice is divisibleby three? 6. When you roll three dice, what is the probabilitythat (a) the three dice sum to five? (b) at least one of the dice is a five? (c) at least one of the dice is an even number? (d) the sum of the three dice is divisibleby three? (e) the three dice show different values? (f) at least two of the dice show the same value? 7. A slot machine has five wheels, each of which has five symbols: an apple, a grape, a peach, a pear, and a plum. When you pull a lever, the wheels roll around and five fruits are randomly displayed. What are the chances to get at least one apple after one pull of the lever? After two pulls? After five pulls? This problem is based on one sent to Ask Marilyn.7

7Ask Marilyn, Parade Magazine,March 14, 2010.

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xxx — #30 i i

xxx How to Read Mathematics

8. When you flip two coins, what is the probabilitythat one coin is heads and the other is tails? 9. When you flip four coins, what is the probability that two coins are heads and the other two are tails? Hint: There are 16 equally likely outcomes. 10. You have two cars and twoidentical keys for each car. All four keys are kept in a box. You choose a key randomly and drive to work in whichever car that key fits. When you return from work you replace the key in the box. Later on, you go out shopping and once again randomly choose a key from the box. What’s the probability that you drive the same car to work and to shopping? What about when you have three cars and six keys? n cars and 2n keys? 11. In tennis, you are allowed two attempts when you serve. If the first attempt misses, you get one more chance. If both attempts miss, you “doublefault” and lose the point. You are a tennis player with a wicked first serve that lands legally only half the time. Your second serve is more conservative and lands legally 9 out of 10 times. What is the probabilitythat you double fault? Hint: Imagine what will typically happen if you try to serve 20 times. 12. I remove two socks from a drawer containing two identical white socks and two iden- tical black socks? What is the probabilitythat I have a matching pair? Hint: Label the socks and count how many ways there are to choose two socks from four socks. 13. I remove two socks from a drawer containing three identical white socks and three identical black socks? What is the probabilitythat I have a matching pair? Hint: Label the socks and count how many ways there are to choose two socks from six socks. 14. I remove two socks from a drawer containing n identical white socks and n identical black socks? What is the probabilitythat I have a matching pair? Hint: Generalize the ideas in the previous two problems. Your answer depends on n. 15. The passage below called Gambler’s Ruin is taken from Stephen Wolfram’s Math- World8 on the internet. Try to read this passage and explore its meaning using the hints and guidelines in this chapter. Explain exactly what probabilities are being cal- culated and justify the formulas given for them. Explain the consequences of this theorem. If you play Poker, try to apply this passage to your chances of winning a tournament when you are the “short stack.”

Let two players each have a finite number of pennies (say, n1 for player one and n2 for player two). Now, flip one of the pennies (from either player), with each player having 50% probability of winning, and transfer a penny from the loser to the winner. Now repeat the process until one player has all the pennies. If the process is repeated indefinitely, the probability that one of the two play- ers will eventually lose all his pennies must be 100%. In fact, the chances P1 and P2 that players one and two, respectively, will be rendered penniless are n n 2 and 1 ; n1 n2 n1 n2 C C 8Weisstein, Eric W. “Gambler’s Ruin.” From MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/GamblersRuin.html .

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xxxi — #31 i i

How to Read Mathematics xxxi

i.e., your chances of going bankrupt are equal to the ratio of pennies your oppo- nent starts out with to the total number of pennies. Therefore, the player starting out with the smallest number of pennies has the greatest chance of going bankrupt. Even with equal odds, the longer you gam- ble, the greater the chance that the player startingout with the most pennies wins. Since casinos have more pennies than their individual patrons, this principle al- lowscasinosto alwayscome out ahead in thelong run.And the common practice of playinggames withodds skewed in favor of thehouse makes this outcome just that much quicker.

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page xxxii — #32 i i

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page 195 — #227 i i

Resources for Rediscovering Mathematics

There are many excellent resources for the person interested in rediscovering mathematics.

Organizations National Association of Math Circles (www.mathcircles.org/)  Mathematical Circles are a form of education enrichment and outreach that bring mathematicians and mathematical scientists into direct contact with pre-college stu- dents. These students, and sometimes their teachers, meet with mathematical profes- sionals in an informal setting, after school or on weekends, to work on interesting problems or topics in mathematics. The goal is to get the students excited about the mathematics, giving them a setting that encourages them to become passionate about mathematics.

– The Math Circle (www.themathcircle.org/), foundedin 1994 by Boband Ellen Kaplan, offers courses designed for boys and girls ages 5 through 17 who enjoy math and want the added challenge of exciting topics that are normally outside the school curriculum. Classes encourage a free discussion of ideas and the atmosphere is friendly and relaxed. – The Math Teachers’ Circle Network (www.mathteacherscircle.org/), sponsored by the American Institute of Mathematics (AIM), promotes the enjoy- ment of problem solving among middle school mathematics teachers.

MATHCOUNTS Foundation (mathcounts.org/)  MATHCOUNTS is a national enrichment, club and competition program that pro- motes middle school mathematics achievement through grassroots involvement in every U.S. state and territory. MATHCOUNTS heightens student interest in math- ematics by making math achievement as challenging, exciting and prestigious as a school sport. MATCHOUNTS promotes problem solving. Spending time discovering solutions to MATHCOUNTS problems is an excellent alternative or supplement to a more tradi- tional mathematics curriculum.

195

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page 196 — #228 i i

196 Resources for Rediscovering Mathematics

MathPath (www.mathpath.org/)  MathPath is a summer experience for talented middle school mathematics students. They offer excellent resources for teachers and students. Although they cater to tal- ented students, their philosophy and methodology can be used for all students, as the right training and teachers can bring out the talent in many students.

– MathCamp (www.mathcamp.org/) is a similar program for high school level students.

The Mathematical Association of America (www.maa.org/)  The MAA, which focuses mainly on mathematics at the college level and above, sponsors the American Mathematics Competitions (www.unl.edu/amc/), offer- ing problem solving opportunities at the middle school and high school levels. The MAA also offers many resources that can be used by creative teachers in the class- room to foster mathematical discovery.

Math for America (www.mathforamerica.org/)  “The MFA is a nonprofit organization with a mission to improve math education in secondary public schools in the United States by recruiting, training, and retaining outstanding mathematics teachers. Competitive four year Master Teaching Fellow- ships offer training, experience, community, and generous stipends.” The MFA is a good place to look for an active community of excellent mathematics educators.

The National Council of Teachers of Mathematics (www.nctm.org/)  The NCTM is committed to the principal that learning mathematics with understand- ing is essential. They offer many resources, publications, and workshops to teach the art of exploring and discovering mathematics for grades K–12.

The Math Museum (momath.org/)  Glen Whitney, a former hedge fund manager, is currently designing a Mathematics Museum in Manhattan. Whitney believes that math is underappreciated and poorly taught. He does not like the linear approach of algebra, geometry, trigonometry, pre- calculus, calculus, and so on. He dreams of a museum where people can puzzle inter- actively over the mathematical landscape.1

Teaching-Focused Internet Sites Math Forum —An online community offering resources to improve mathematical ed-  ucation. Includes the Internet Math Library—an indexed compilation of mathematical topics and lesson plans. mathforum.org/

Math is Fun—Fun puzzles and lessons on elementary school level topics.  www.mathsisfun.com/

1“Math-hattan” by Nick Paumgarten, The New Yorker, August, 2009.

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page 197 — #229 i i

Resources for Rediscovering Mathematics 197

Brain Pop—A wide variety of interactive lessons for elementary school.  www.brainpop.com/math/seeall/

Cynthia Lanius’ Lessons—A fun and eclectic variety of lessons at the middle school  level. math.rice.edu/lanius/Lessons/ Thinking Mathematics!—This site by James Tanton offers nice resources for students  and teachers to discover, explore, and investigate authentic mathematical questions at the high school level. www.jamestanton.com/ Shackleford’s Puzzle Collection—Agreat place to find or contributea new math puz-  zle. mathproblems.info/ Larry Copes Ed-Math—A simulator for math teachers to try out lesson plans.  edmath.org/

Math-ManiCS—Lesson plans for discrete math topics.  www.mathmaniacs.org/lessons/index.html

Content-Focused Internet Sites Cut The Knot—A comprehensive collection of mathematical topics presented inter-  actively. www.cut-the-knot.org/Curriculum/index.shtml Art of Problem Solving—An online community for problem solving resources.  www.artofproblemsolving.com/

Jim Loy’s Math Pages—A low tech but high quality collection of interesting mathe-  matical topics. www.jimloy.com/math/math.htm Numericana—Another low tech but high quality collection of mathematical topics.  www.numericana.com/

World of Mathematics—A sophisticated and comprehensive reference.  mathworld.wolfram.com/

Mathnerds—Online expert guidance in math. www.mathnerds.com/  Mu Alpha Theta Honor Society—Lots of useful links.  www.mualphatheta.org/Problem Corner/Links.aspx

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page 198 — #230 i i

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page 199 — #231 i i

Further Reading

A number of excellent resources are referenced in the footnotesof this book. The following additional books, listed in order of most recent publication date, provide a starting place for further reading. The Heart of Mathematics: An Invitation to Effective Thinking, 3rd edition by Edward  Burger and Michael Starbird, Wiley and Sons, 2010. This book claims that the beauty and applicabilityof mathematics isoften hidden from students. The book provides many good examples of mathematics and opportunities for discovery. A Mathematicians Lament by Paul Lockhart, Bellevue Literary Press, 2009.  This short book is a critiqueof current mathematics educationintheUS that will open your eyes to alternatives. A Gardner’s Workout: Training the Mind and Entertaining the Spirit by Martin Gard-  ner, AK Peters, 2001. Anythingby MartinGardner goes a longway toinspiringa new generation of teachers and students with a passion for mathematics and problem solving. What is Mathematics? An Elementary Approach to Ideas and Methods, by Richard  Courant and Herbert Robbins, revised by Ian Stewart, Oxford University Press, 1996. This revision of a classic, first published in 1941, is a comprehensive tour of math- ematics for the serious enthusiast. This is the book I recommend to people who are looking for the real thing, and fancy themselves unafraid of a challenge. Mathematics, A Human Endeavor, 3rd edition by Harold R. Jacobs, WH Freeman,  1994. This is a well-organized collection of mathematics for grades K–12, outside the usual topics and curricula. Journey Through Genius by William Dunham, Penguin, 1991.  This book offers a guided tour of an eclectic collection of famous mathematical dis- coveries.

199

i i

i i i i “master˙color” — 2011/6/30 — 12:23 — page 200 — #232 i i

200 Resources for Rediscovering Mathematics

How to Solve It by G. P´olya, Princeton University Press, 1988.  This is a classic book about problem solving and how to do mathematics.

i i

i i i i “master˙color” — 2016/2/22 — 11:59 — page 201 — #233 i i

Index

acceleration of falling objects, 67–69 The Art of Calculation (ben Gershon), xix–xx active reading, xviii–xix ASCII (American Standard Code for Informa- Additive Law of Exponents, 190 tion Interchange), 110 Albert, Abbot, 93 averages, 77–92 algorithms, 93–111 arithmetic, 77, 82–85, 87–88 brute force, 96, 99, 100–101, 103–104 centroidal, 89 challenges, 109–111 challenges, 89–92 cryptography and, 106–108 contraharmonic, 89 e-commerce and, 105 exploring relationships between, 82–89 efficiency or speed of, 99–101 geometric, 79–82, 83–84, 86–88 Egyptian multiplication algorithm, 101–103 harmonic, 78–81, 83–86, 88 Euclid’s algorithm, 96–98 heronian, 89 factoring, 99 judgment in using calculators for, 13–14 fast modular exponentiation algorithm, 103– root-mean-square, 89 105 axioms, 35 greatest common divisor, 99 Babylonians, and Pythagorean triples, 24 overview, 93–94 Babylonian version of quadratic equation, 135, RSA algorithm, 108–109, 119 139 third grade multiplication, 101–102 basketball, 66–69 two jug puzzle, 94–96 Benjamin, Art, 18–20 American Standard Code for Information In- Biblical calculation of pi, 170–173 terchange (ASCII), 110 binomials, 130 applications of mathematics.See real world uses birthday paradox, xxii–xxviii of mathematics brick weight puzzle, 152–153 area. See also pi and area brute force algorithm, 96, 99, 100–101, 103– of circle, 173–175 104 of cone, 180 of ellipse, 181 Caesar’s cipher, 106 of equilateral triangle, 119–120 calculators, judgment in using, 13–29 four quarter-circles inscribed in a square, averages and, 13–14 175–179 challenges, 26–29 of rectangle, 169, 191–192 clever calculations, 20–22 of trapezoid, 191–193 distributive law of multiplication over ad- of triangle, 191 dition/subtraction, 15–17 arithmetic average, 77, 82–85, 87–88 hand trick for multiplying by nine, 14–15, arithmetic mean, 83 15f

201

i i

i i i i “master˙color” — 2016/2/22 — 11:59 — page 202 — #234 i i

202 Index

Pythagorean triples, 22–25 craps, 146–151, 152 three-digit square trick, 18–20 Crazy Craps, 153 two-digit square trick, 18–20 credit card encryption, 93, 99 word problems and, 14 cryptography, 93, 106–109 card trick, probability, 155–156 cube root, 20, 70 careless casino, 151–152 cubes, 161, 162f, 164 carnival dice game, 143–145 cubits, 171–173 ”casting out nines,” 28 Cauchy, 167 decimals, terminating vs. repeating, 1–10 centroidal average, 89 decryption, 105–109 challenges Descartes, Ren´e, 161 averages, 89–92 Descartes-Euler polyhedral formula, 160–162 Gambler’s Ruin, xxx diamond numbers, 43 games and gambling, 152–157 dice games geometric series, 38–40 carnival, 143–145 hand calculation tricks, 26–29 craps, 146–151 infinite series, 38–40 Crazy Craps, 153 irrational numbers, 11 Die Hard: With a Vengeance, 93 mental calculation tricks, 26–29 discovery, mathematical. See mathematical dis- pi and area, 179–183 covery probability, xxix–xxxi discriminant, 133 Pythagoras’ theorem, 124–127 distributive law of multiplication over addition/ Pythagorean triples, 26–27 subtraction, 15–17, 130 rational numbers, 10 divergent series, 39–40 reading protocols, xxix–xxxi dodecahedrons, 162f, 164, 166 real world uses of mathematics, 72–75 Dunham, Bill, 160 repeating and terminating decimals, 10–11 solving equations, 140–142 e-commerce applications for algorithms, 105 ciphers, 106 effective teaching, 9–10 circle, area of, 173–175 Egyptian multiplication algorithm, 101–103 circumference, 169 elegant counting, 43 classroom interaction, 185–193 The Elements (Euclid), 35–38, 98, 162 emphasizing understanding, 188–193 Eliot, T. S., xxii three- or four-digit number challenge dia- Elkies, Noam, 25 logue, 185–188 ellipse, area of, 181 closed form formula, 59 Emblems of Mind (Rothstein), xvii coin flipping, and probability, 154, 155 encryption, 93, 99, 103, 105–109 completing the square, 137, 140 equilateral triangles, 162–163, 175–176 conditions, necessary vs. sufficient, 3–5 Euclid, 35–38, 93, 162 conductor, math teacher as, 2–9 Euclid’s algorithm, 96–98, 99, 101, 103, 105, cone, area of, 180 107–109 contraharmonic average, 89 Euclid’s proof of Pythagoras’ theorem, 113– convergent series, 39–40 115 counting incorrectly, 159–160, 163 Euclid’s Propositions, 125–126 counting pairs, 159–160 Euler, Leonhard, 160, 161f counting tricks, 159–167 Euler’s formula, 160–162, 163–164, 166 challenges, 166–167 Euler: The Master of Us All (Dunham), 160 counting incorrectly, 159–160, 161, 163 Eve, Howard, 88 counting pairs, 159–160 counting sides of polyhedra, 163–164 factoring algorithm, 99, 100–101 Euler’s formula, 160–162, 163–164, 166 false equations, 20–21, 25 Platonic solids, 162–163, 162f fast modularexponentiation algorithm, 103–105 soccer ball puzzle, 160, 164–166 Fermat, Pierre de, 22, 24, 108

i i

i i i i “master˙color” — 2016/2/22 — 11:59 — page 203 — #235 i i

Index 203

Fermat near-misses, 25 Jacob’s staff, 44 Fermat’s Last Theorem, 22, 24 Jordan, Michael, 66–67 Fermat’s Little Theorem, 108 Julius Caesar, 106 figurate numbers, 61, 62 flipping coins, and probability, 154, 155 knock hockey, 64–66 FOIL (First, Outer, Inner, Last) rule, 129–131 four quarter-circles inscribed in a square, area Lakatos, Imre, 161 calculation, 175–178f, 175–179 Lam`e, Gabrielle, 101 Levi ben Gershon, xix–xx, 44–52, 171, 179, Gambler’s Ruin challenge, xxx–xxxi 180 gambling. See games and gambling Levi ben Gershon’s Propositions, 48–52 games and gambling, 143–157 Levi ben Gershon’s sums, 44–52 careless casino, 151–152 linear rate of growth, 69 carnival dice game, 143–145 Loomis, E. S., 118 challenges, 152–157 Loyd, Sam, 152–153 craps, 146–151, 152 Gauss, Carl Friedrich, 42, 45 Maaseh Hoshev (The Art of Calculation) (Levi geometric average, 79–82, 83–84, 86–88 ben Gershon), xix–xx, 45 geometric mean, 83, 85, 87 math by pictures. See Pythagoras’ theorem geometric series, 33, 36–38 “mathemagician,” 18 Gersonides. See Levi ben Gershon mathematical discovery, 1–11 Google, 93 challenges, 10–11 greatest common divisor, 97–98, 99 effective teaching, 9–10 greatest common measure. See greatest com- math teacher as conductor, 2–9 mon divisor terminating vs. repeating decimals, 1–10 grids, squares and rectangles in, 52–54 mathematical induction, 19, 51–52 guessing, importance of, 3 mathematical literacy, 188 guessing game probability challenge, 156 mathematical maturity, xxii mathematical proofs, 35–38 handbreadth, 171–173 mathematical thinking, 41–62 hand trick for multiplying by nine, 14–15 challenges, 61–62 hang-time, 66–67 diamond numbers, 75 Hardy, G. H., 63 grids, squares and rectangles in, 52–54 harmonic average, 78–81, 83–86, 88 Levi ben Gershon’s sums, 44–52 harmonic mean, 83 square numbers, 41–42 heronian average, 89 triangle grid puzzle, 55–60 Heron’s formula, 179–180 triangular numbers, 42–43 hexagonal numbers, 61 A Mathematician’s Apology (Hardy), 63 hexagons, 160 mathematics, reading protocol for. See reading Hippasus, 8–9, 11 protocol for mathematics Homer Simpson’s false equation, 20–21, 25 mathematics, real world uses of. See real world hypotenuse, 113 uses of mathematics math teacher as conductor, 2–9 Ibn Ezra, Abraham ben Meir, 141–142 Math-World (Wolfram), xxx–xxxi icosahedrons, 162f, 163–164 mean, 83, 85, 87 “if and only if,” 4 median, 81 independent events, 147–148 memorizing vs. understanding, 129–142 induction, mathematical, 19, 51–52 challenges, 140–142 infinite series, 31–34, 36–38 FOIL (First, Outer, Inner, Last) rule, 129– irrational numbers, 10, 169 131 isosceles triangles, 123, 124–125, 127 quadratic equation, 133–137 quadratic equation, real life example of, 138 Jackson, Samuel L., 93 –140

i i

i i i i “master˙color” — 2016/2/22 — 11:59 — page 204 — #236 i i

204 Index

square roots, 131–133 pyramids mental math, 18–20 square-based, 122–124 multiplication principle of probability, 147 triangular-based, 120–122 Pythagoras’ theorem, 113–127 near-misses, equations as, 25 author’s proof of, 115–120 necessary conditions, 3–5 challenges, 124–127 Nelsen, Roger, 124 Euclid’s proof of, 113–115 nine height of square-based pyramid, 122–124 hand trick for multiplying by, 14–15 height of triangular-based pyramid, 120– numbers divisible by, 28, 187–188 122 nodes, on polyhedron, 160–161, 163–164, 166 see-through proof of, 114–115 non-primitive Pythagorean triples, 23 Pythagorean triples, 22–25 nth square pyramid number, 62 nth triangular pyramid number, 62 quadratic equation, 133–137, 138–140

octahedrons, 162f, 164 rates of growth, 69–72 O’Neal, Shaquille, 148 rational numbers, 10 reading protocol for mathematics, xvii–xxxi pairs, counting, 159–160 the big picture, understanding, xviii Parade Magazine, 34 birthday paradox, xxii–xxviii paradoxes of motion. See Zeno’s paradoxes of challenges, xxix–xxxi motion mathematical maturity, xxii participatory reading, xviii–xix owning the idea, xxi passive reading, xviii–xix precise vocabulary, importance of, xxi–xxii pattern matching, 134–135, 190 reading actively, xviii–xix pentagonal numbers, 61–62 reading with concentration, xix–xx pentagons, 160, 162–166 real world uses of mathematics, 63–75 pi and area, 169–183 basketball, 66–69 area of a circle, 173–175 challenges, 72–75 Biblical calculation of pi, 170–173 knock hockey, 64–66 challenges, 179–183 overview, 63–64 four quarter-circles inscribed in a square, rates of growth, 69–72 area calculation, 175–178f, 175–179 rectangles, 16–17, 52–55, 169, 173–174, 191– overview, 169–170 192 pizza pie techniquefor measuring area of a cir- repeating decimals, 1–10 cle, 173 right triangles. See Pythagoras’ theorem Platonic solids, 162–163, 162f Rivest, Ron, 119 Plato’s formula for Pythagorean triples, 23–24 root-mean-square average, 89 Plimpton 322 Tablet, 24f Rothstein, Ed, xvii Poisson, Simeon, 94 RSA algorithm, 108–109, 119 polygons, 160–163, 182 polyhedra, 160–164 Scrabble, 155 prime numbers, 3 Sea of Solomon, 170, 170f, 179, 180 primitive Pythagorean triples, 23 secure e-commerce transactions, 105 probability for an event not to occur, 148 see-through proof of Pythagoras’theorem, 114 probability theory, 146–152 –115 proofs, 4, 35–38 series, convergent and divergent, 39–40 Proofs that Really Count (Benjamin and Quinn), “shift and subtract” method for geometric se- 43 ries, 33, 39–40 “proofs without words,” 124 similar triangles, 125 Proposition IX-35, Euclid, 36–38, 40 The Simpsons and Homer’s false equation, 20– Propositions, Levi’s, 48–52, 125–126 21, 25 public key cryptography, 107–108 soccer ball puzzle, 160, 164–166

i i

i i i i “master˙color” — 2016/2/22 — 11:59 — page 205 — #237 i i

Index 205

A Song for Simeon (Eliot), xxii unit conversion, 34–35 spontaneous interaction, 185 uses of mathematics. See real world uses of square-based pyramids, 122–124 mathematics square numbers, 41–42, 47 square roots, 131–133 Vigen`ere, Blaise de, 106 substitution ciphers, 106 Vigen`ere cipher, 106 sufficient conditions, 3–5 Vinci, Leonardo da, 172 sums of consecutive numbers, 44–52 Vitruvian Man, 172, 172f symmetry, proof by, 144 vocabulary, mathematical, xxi–xxii vos Savant, Marilyn, 34–35 tangent quadrilaterals, 126 terminating decimals, 1–10 weight of a brick puzzle, 152–153 tetrahedrons,120–122,126–127,161,162f, 164 Wiles, Andrew, 24–25 theorems, 35 Willis, Bruce, 93 third grade multiplication algorithm, 101–102 Wolfram, Stephen, xxx–xxxi three-digit square trick, 18–20 word problems, 14, 189 trapezoids, 191–193 triangle grid puzzle, 55–60 Yao Ming, 69 triangles equilateral, 162–163, 175–176 Zeno’s paradoxes of motion, 31–40 isosceles, 123, 124–125, 127 challenges, 38–40 similar, 125 convergent and divergent series, 32, 39–40 triangular-based pyramids, 120–122 infinite series, 31–34 triangular numbers, 42–43, 46–47 proofs, 35–38 trinomials, 130 unit conversion, 34–35 Turing, Alan, 106 two-digit square trick, 18–20 two jug puzzle, 93, 94–96

i i

i i i i “master˙color” — 2016/2/22 — 11:59 — page 206 — #238 i i

i i

i i i i “master˙color” — 2016/2/22 — 11:59 — page 207 — #239 i i

About the Author

Shai Simonson is a professor of computer science at Stonehill College. Simonson was born in Buffalo, grew up in New York City, spent ten years in Chicago, and now makes his home in the Boston area. At various times, he has taught gym, science, mathematics, and computer science to students from first grade through graduate school. Simonson is passionate about teaching mathematics in an active, experiential, and dynamic fashion. He plays go and bridge, dabbles with poker and Scrabble, loves to hike, cycle, bowl, sing, and play disc-golf. He is married with three children. Simonson earned his B.A. in mathematics at Columbia University, and M.S. and Ph.D. degrees in computer science at Northwestern University. He is a member of the MAA and the ACM. He spent two years (1983 and 1999) teaching and doing research in Israel. The experiences in 1999 are chronicled in an on-line journal (web.stonehill.edu/ compsci/shai.htm). Simonson has taught mathematics and computer science courses for almost 30 years at a number of colleges and universities including University of Illinois, Northwestern, Tel Aviv University, ArsDigita University, and Stonehill College. He also taught at the middle school and high school levels in New York, Chicago, and Boston. For seven years at the South Area Solomon Schechter Day School in Norwood, MA, Simonson designed and tested a curriculum for middle school students, based on the principles he presents in Rediscovering Mathematics.Simonson uses his book as the text in the mathematics course for education majors at Stonehill College, a course required by the state of Massachusetts. Simonson has published articles in mathematics education, history of mathematics, popular mathematics, theoretical computer science, and computer science education. He has lectured all over the world. At the height of the dot-com boom in 2000, Simonson directed a corporate-sponsored one-year post-baccalaureate computer science program in Cambridge, MA called ArsDigita University. Videotaped lectures of his mathematics, al- gorithms, and theory of computation courses, located at the university’s website, are used all over the world by thousands of people (aduni.org), especially students in emerging countries. Simonson was awarded a number of grants and awards from the National Sci- ence Foundation for a variety of projects, both pedagogical and scholarly, including one award with Ralph Bravaco that led to the writing of his first book, Java Programming: From the Ground Up, McGraw Hill, 2009.

207

i i

i i