to Inspire Teachers and Students Alfred S.Posamentier to Inspire Teachers and Students Alfred S.Posamentier Association for Supervision and Curriculum Development Alexandria, Virginia USA AssociationforSupervisionandCurriculumDevelopment 1703N.BeauregardSt.*Alexandria,VA22311-1714USA Telephone:800-933-2723or703-578-9600*Fax:703-575-5400 Website:http://www.ascd.org*E-mail:[email protected] GeneR.Carter,ExecutiveDirector;NancyModrak,DirectorofPublishing;JulieHoutz,DirectorofBook Editing&Production;DarcieRussell,ProjectManager;TechnicalTypesetting,Inc.,Typesetting;Tracey Smith,Production Copyright©2003byAlfredS.Posamentier.Allrightsreserved.Nopartofthispublicationmaybe reproducedortransmittedinanyformorbyanymeans,electronicormechanical,includingphotocopy, recording,oranyinformationstorageandretrievalsystem,withoutpermissionfromASCD.Readerswho wishtoduplicatematerialmaydosoforasmallfeebycontactingtheCopyrightClearanceCenter(CCC), 222RosewoodDr.,Danvers,MA01923,USA(telephone:978-750-8400;fax:978-750-4470; Web:http://www.copyright.com).ASCDhasauthorizedtheCCCtocollectsuchfeesonitsbehalf.Requests toreprintratherthanphotocopyshouldbedirectedtoASCD’spermissionsofficeat703-578-9600. Coverartcopyright©2003byASCD.CoverdesignbyShelleyYoung. ASCDpublicationspresentavarietyofviewpoints.Theviewsexpressedorimpliedinthisbookshouldnot beinterpretedasofficialpositionsoftheAssociation. AllWeblinksinthisbookarecorrectasofthepublicationdatebelowbutmayhavebecomeinactiveor otherwisemodifiedsincethattime.Ifyounoticeadeactivatedorchangedlink,pleasee-mail [email protected]“LinkUpdate”inthesubjectline.Inyourmessage,pleasespecifytheWeb link,thebooktitle,andthepagenumberonwhichthelinkappears. netLibrary E-Book : ISBN 0-87120-852-0 Price: $27.95 Quality Paperback : ISBN 0-87120-775-3 ASCD Product No. 103010 ASCD Member Price: $22.95 nonmember Price: $27.95 LibraryofCongressCataloging-in-PublicationData (for paperback book) Posamentier,AlfredS. Mathwonderstoinspireteachersandstudents/[AlfredS. Posamentier]. p.cm. Includesbibliographicalreferencesandindex. ISBN0-87120-775-3(alk.paper) 1.Mathematics–Studyandteaching.2.Mathematicalrecreations.I. Title. QA11.2.P642003 510—dc21 2003000738 In memory of my beloved parents, who, after having faced monumental adversities, provided me with the guidance to develop a love for mathematics, and chiefly to Barbara, without whose support and encouragement this book would not have been possible. Math Wonders to Inspire Teachers and Students Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Chapter 1 The Beauty in Numbers . . . . . . . . . . . . . . . . . . 1 1.1 Surprising Number Patterns I . . . . . . . . . . . . . . . 2 1.2 Surprising Number Patterns II . . . . . . . . . . . . . . . 5 1.3 Surprising Number Patterns III . . . . . . . . . . . . . . 6 1.4 Surprising Number Patterns IV . . . . . . . . . . . . . . 7 1.5 Surprising Number Patterns V . . . . . . . . . . . . . . . 9 1.6 Surprising Number Patterns VI . . . . . . . . . . . . . . 10 1.7 Amazing Power Relationships . . . . . . . . . . . . . . . 10 1.8 Beautiful Number Relationships. . . . . . . . . . . . . . 12 1.9 Unusual Number Relationships . . . . . . . . . . . . . . 13 1.10 Strange Equalities . . . . . . . . . . . . . . . . . . . . . . 14 1.11 The Amazing Number 1,089 . . . . . . . . . . . . . . . 15 1.12 The Irrepressible Number 1 . . . . . . . . . . . . . . . . 20 1.13 Perfect Numbers . . . . . . . . . . . . . . . . . . . . . . 22 1.14 Friendly Numbers . . . . . . . . . . . . . . . . . . . . . . 24 1.15 Another Friendly Pair of Numbers . . . . . . . . . . . . 26 1.16 Palindromic Numbers . . . . . . . . . . . . . . . . . . . . 26 1.17 Fun with Figurate Numbers . . . . . . . . . . . . . . . . 29 1.18 The Fabulous Fibonacci Numbers. . . . . . . . . . . . . 32 1.19 Getting into an Endless Loop . . . . . . . . . . . . . . . 35 1.20 A Power Loop . . . . . . . . . . . . . . . . . . . . . . . . 36 1.21 A Factorial Loop .√. . . . . . . . . . . . . . . . . . . . . 39 1.22 The Irrationality of 2 . . . . . . . . . . . . . . . . . . . 41 1.23 Sums of Consecutive Integers . . . . . . . . . . . . . . . 44 Chapter 2 Some Arithmetic Marvels . . . . . . . . . . . . . . . . . 47 2.1 Multiplying by 11 . . . . . . . . . . . . . . . . . . . . . . 48 2.2 When Is a Number Divisible by 11? . . . . . . . . . . . 49 v 2.3 When Is a Number Divisible by 3 or 9? . . . . . . . . . 51 2.4 Divisibility by Prime Numbers . . . . . . . . . . . . . . 52 2.5 The Russian Peasant’s Method of Multiplication . . . . 57 2.6 Speed Multiplying by 21, 31, and 41 . . . . . . . . . . . 59 2.7 Clever Addition . . . . . . . . . . . . . . . . . . . . . . . 60 2.8 Alphametics . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.9 Howlers . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.10 The Unusual Number 9 . . . . . . . . . . . . . . . . . . 69 2.11 Successive Percentages . . . . . . . . . . . . . . . . . . . 72 2.12 Are Averages Averages? . . . . . . . . . . . . . . . . . . 74 2.13 The Rule of 72 . . . . . . . . . . . . . . . . . . . . . . . 75 2.14 Extracting a Square Root . . . . . . . . . . . . . . . . . 77 Chapter 3 Problems with Surprising Solutions . . . . . . . . . . . . 79 3.1 Thoughtful Reasoning . . . . . . . . . . . . . . . . . . . 80 3.2 Surprising Solution . . . . . . . . . . . . . . . . . . . . . 81 3.3 A Juicy Problem . . . . . . . . . . . . . . . . . . . . . . 82 3.4 Working Backward . . . . . . . . . . . . . . . . . . . . . 84 3.5 Logical Thinking . . . . . . . . . . . . . . . . . . . . . . 85 3.6 It’s Just How You Organize the Data . . . . . . . . . . . 86 3.7 Focusing on the Right Information . . . . . . . . . . . . 88 3.8 The Pigeonhole Principle . . . . . . . . . . . . . . . . . 89 3.9 The Flight of the Bumblebee . . . . . . . . . . . . . . . 90 3.10 Relating Concentric Circles . . . . . . . . . . . . . . . . 92 3.11 Don’t Overlook the Obvious . . . . . . . . . . . . . . . . 93 3.12 Deceptively Difficult (Easy) . . . . . . . . . . . . . . . . 95 3.13 The Worst Case Scenario . . . . . . . . . . . . . . . . . 97 Chapter 4 Algebraic Entertainments . . . . . . . . . . . . . . . . . . 98 4.1 Using Algebra to Establish Arithmetic Shortcuts . . . . 99 4.2 The Mysterious Number 22 . . . . . . . . . . . . . . . . 100 4.3 Justifying an Oddity . . . . . . . . . . . . . . . . . . . . 101 4.4 Using Algebra for Number Theory . . . . . . . . . . . . 103 4.5 Finding Patterns Among Figurate Numbers . . . . . . . 104 4.6 Using a Pattern to Find the Sum of a Series . . . . . . 108 4.7 Geometric View of Algebra . . . . . . . . . . . . . . . . 109 4.8 Some Algebra of the Golden Section . . . . . . . . . . . 112 vi 4.9 When Algebra Is Not Helpful . . . . . . . . . . . . . . . 115 4.10 Rationalizing a Denominator . . . . . . . . . . . . . . . 116 4.11 Pythagorean Triples . . . . . . . . . . . . . . . . . . . . . 117 Chapter 5 Geometric Wonders . . . . . . . . . . . . . . . . . . . . . 123 5.1 Angle Sum of a Triangle . . . . . . . . . . . . . . . . . . 124 5.2 Pentagram Angles . . . . . . . . . . . . . . . . . . . . . . 126 5.3 Some Mind-Bogglers on (cid:1) . . . . . . . . . . . . . . . . 131 5.4 The Ever-Present Parallelogram . . . . . . . . . . . . . . 133 5.5 Comparing Areas and Perimeters . . . . . . . . . . . . . 137 5.6 How Eratosthenes Measured the Earth . . . . . . . . . . 139 5.7 Surprising Rope Around the Earth . . . . . . . . . . . . 141 5.8 Lunes and Triangles . . . . . . . . . . . . . . . . . . . . 143 5.9 The Ever-Present Equilateral Triangle . . . . . . . . . . 146 5.10 Napoleon’s Theorem . . . . . . . . . . . . . . . . . . . . 149 5.11 The Golden Rectangle . . . . . . . . . . . . . . . . . . . 153 5.12 The Golden Section Constructed by Paper Folding . . . 158 5.13 The Regular Pentagon That Isn’t . . . . . . . . . . . . . 161 5.14 Pappus’s Invariant . . . . . . . . . . . . . . . . . . . . . . 163 5.15 Pascal’s Invariant . . . . . . . . . . . . . . . . . . . . . . 165 5.16 Brianchon’s Ingenius Extension of Pascal’s Idea . . . . 168 5.17 A Simple Proof of the Pythagorean Theorem . . . . . . 170 5.18 Folding the Pythagorean Theorem . . . . . . . . . . . . 172 5.19 President Garfield’s Contribution to Mathematics . . . . 174 5.20 What Is the Area of a Circle? . . . . . . . . . . . . . . . 176 5.21 A Unique Placement of Two Triangles . . . . . . . . . . 178 5.22 A Point of Invariant Distance in an Equilateral Triangle . . . . . . . . . . . . . . . . 180 5.23 The Nine-Point Circle . . . . . . . . . . . . . . . . . . . 183 5.24 Simson’s Invariant . . . . . . . . . . . . . . . . . . . . . 187 5.25 Ceva’s Very Helpful Relationship . . . . . . . . . . . . . 189 5.26 An Obvious Concurrency? . . . . . . . . . . . . . . . . . 193 5.27 Euler’s Polyhedra . . . . . . . . . . . . . . . . . . . . . . 195 Chapter 6 Mathematical Paradoxes . . . . . . . . . . . . . . . . . . 198 6.1 Are All Numbers Equal? . . . . . . . . . . . . . . . . . . 199 6.2 −1 Is Not Equal to +1 . . . . . . . . . . . . . . . . . . . 200 vii 6.3 Thou Shalt Not Divide by 0 . . . . . . . . . . . . . . . . 201 6.4 All Triangles Are Isosceles . . . . . . . . . . . . . . . . 202 6.5 An Infinite-Series Fallacy . . . . . . . . . . . . . . . . . 206 6.6 The Deceptive Border . . . . . . . . . . . . . . . . . . . 208 6.7 Puzzling Paradoxes . . . . . . . . . . . . . . . . . . . . . 210 6.8 A Trigonometric Fallacy . . . . . . . . . . . . . . . . . . 211 6.9 Limits with Understanding . . . . . . . . . . . . . . . . . 213 Chapter 7 Counting and Probability . . . . . . . . . . . . . . . . . . 215 7.1 Friday the 13th! . . . . . . . . . . . . . . . . . . . . . . . 216 7.2 Think Before Counting . . . . . . . . . . . . . . . . . . . 217 7.3 The Worthless Increase . . . . . . . . . . . . . . . . . . . 219 7.4 Birthday Matches . . . . . . . . . . . . . . . . . . . . . . 220 7.5 Calendar Peculiarities . . . . . . . . . . . . . . . . . . . . 223 7.6 The Monty Hall Problem . . . . . . . . . . . . . . . . . 224 7.7 Anticipating Heads and Tails . . . . . . . . . . . . . . . 228 Chapter 8 Mathematical Potpourri . . . . . . . . . . . . . . . . . . . 229 8.1 Perfection in Mathematics . . . . . . . . . . . . . . . . . 230 8.2 The Beautiful Magic Square . . . . . . . . . . . . . . . . 232 8.3 Unsolved Problems . . . . . . . . . . . . . . . . . . . . . 236 8.4 An Unexpected Result . . . . . . . . . . . . . . . . . . . 239 8.5 Mathematics in Nature . . . . . . . . . . . . . . . . . . . 241 8.6 The Hands of a Clock . . . . . . . . . . . . . . . . . . . 247 8.7 Where in the World Are You? . . . . . . . . . . . . . . . 251 8.8 Crossing the Bridges . . . . . . . . . . . . . . . . . . . . 253 8.9 The Most Misunderstood Average . . . . . . . . . . . . 256 8.10 The Pascal Triangle . . . . . . . . . . . . . . . . . . . . . 259 8.11 It’s All Relative . . . . . . . . . . . . . . . . . . . . . . . 263 8.12 Generalizations Require Proof . . . . . . . . . . . . . . . 264 8.13 A Beautiful Curve . . . . . . . . . . . . . . . . . . . . . 265 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 viii Foreword Bertrand Russell once wrote, “Mathematics possesses not only truth but supremebeauty,abeautycoldandaustere,likethatofsculpture,sublimely pure and capable of a stern perfection, such as only the greatest art can show.” Can this be the same Russell who, together with Alfred Whitehead, authored the monumental Principia Mathematica, which can by no means be regarded as a work of art, much less as sublimely beautiful? So what are we to believe? Let me begin by saying that I agree completely with Russell’s statement, which I first read some years ago. However, I had independently arrived at the same conviction decades earlier when, as a 10- or 12-year-old, I first learned of the existence of the Platonic solids (these are perfectly symmetric three-dimensional figures, called polyhedra, where all faces, edges, and angles are the same—there are five such). I had been reading a book on recreational mathematics, which contained not only pictures of the five Platonic solids, but patterns that made possible the easy construc- tion of these polyhedra. These pictures made a profound impression on me; I could not rest until I had constructed cardboard models of all five. This was my introduction to mathematics. The Platonic solids are, in fact, sublimely beautiful (as Russell would say) and, at the same time, the sym- metries they embody have important implications for mathematics with consequences for both geometry and algebra. In a very real sense, then, they may be regarded as providing a connecting link between geometry and algebra. Although I cannot possibly claim to have understood the full significance of this relationship some 7 decades ago, I believe it fair to say that this initial encounter inspired my subsequent 70-year love affair with mathematics. ix