| About the Author |
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v | |
| Preface |
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xvi | |
| Introduction |
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1 | (4) |
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Cross-Catalog of Enrichment Units |
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5 | |
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Unit 1 Constructing Odd-Order Magic Squares |
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12 | (4) |
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Unit 2 Constructing Even-Order Magic Squares |
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16 | (5) |
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Unit 3 Introduction to Alphametics |
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21 | (5) |
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Unit 4 A Checkerboard Calculator |
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26 | (5) |
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31 | (3) |
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Unit 6 The Tower of Hanoi |
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34 | (4) |
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Unit 7 What Day of the Week Was It? |
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38 | (10) |
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Unit 8 Palindromic Numbers |
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48 | (3) |
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Unit 9 The Fascinating Number Nine |
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51 | (6) |
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Unit 10 Unusual Number Properties |
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57 | (5) |
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Unit 11 Enrichment with a Calculator |
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62 | (4) |
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Unit 12 Symmetric Multiplication |
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66 | (5) |
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Unit 13 Variations on a Theme -- Multiplication |
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71 | (6) |
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Unit 14 Ancient Egyptian Arithmetic |
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77 | (4) |
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81 | (2) |
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83 | (2) |
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Unit 17 Successive Discounts and Increases |
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85 | (4) |
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Unit 18 Prime and Composite Factors of a Whole Number |
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89 | (4) |
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Unit 19 Prime Numeration System |
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93 | (4) |
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Unit 20 Repeating Decimal Expansions |
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97 | (4) |
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Unit 21 Peculiarities of Perfect Repeating Decimals |
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101 | (3) |
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Unit 22 Patterns in Mathematics |
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104 | (4) |
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Unit 23 Googol and Googolplex |
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108 | (4) |
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Unit 24 Mathematics of Life Insurance |
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112 | (3) |
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Unit 25 Geometric Dissections |
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115 | (4) |
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119 | (3) |
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Unit 27 The Four-Color Map Problem |
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122 | (4) |
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Unit 28 Mathematics on a Bicycle |
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126 | (5) |
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Unit 29 Mathematics and Music |
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131 | (4) |
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Unit 30 Mathematics in Nature |
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135 | (6) |
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Unit 31 The Birthday Problem |
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141 | (3) |
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Unit 32 The Structure of the Number System |
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144 | (4) |
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Unit 33 Excursions in Number Bases |
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148 | (3) |
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151 | (4) |
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Unit 35 Reflexive, Symmetric, and Transitive Relations |
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155 | (4) |
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Unit 36 Bypassing an Inaccessible Region |
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159 | (4) |
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Unit 37 The Inaccessible Angle |
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163 | (3) |
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Unit 38 Triangle Constructions |
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166 | (5) |
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Unit 39 The Criterion of Constructibility |
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171 | (5) |
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Unit 40 Constructing Radical Lengths |
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176 | (2) |
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Unit 41 Constructing a Pentagon |
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178 | (3) |
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Unit 42 Investigating the Isosceles Triangle Fallacy |
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181 | (4) |
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Unit 43 The Equiangular Point |
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185 | (4) |
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Unit 44 The Minimum-Distance Point of a Triangle |
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189 | (4) |
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Unit 45 The Isosceles Triangle Revisited |
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193 | (4) |
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Unit 46 Reflective Properties of the Plane |
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197 | (5) |
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Unit 47 Finding the Length of a Cevian of a Triangle |
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202 | (5) |
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Unit 48 A Surprising Challenge |
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207 | (3) |
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Unit 49 Making Discoveries in Mathematics |
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210 | (2) |
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212 | (4) |
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Unit 51 Introducing the Pythagorean Theorem |
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216 | (7) |
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Unit 52 Trisection Revisited |
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223 | (4) |
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Unit 53 Proving Lines Concurrent |
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227 | (5) |
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232 | (2) |
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Unit 55 Proving Points Collinear |
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234 | (4) |
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Unit 56 Angle Measurement with a Circle |
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238 | (4) |
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Unit 57 Trisecting a Circle |
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242 | (4) |
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Unit 58 Ptolemy's Theorem |
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246 | (4) |
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250 | (4) |
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254 | (4) |
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Unit 61 The Nine-Point Circle |
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258 | (3) |
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261 | (3) |
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264 | (4) |
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Unit 64 The Butterfly Problem |
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268 | (4) |
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272 | (5) |
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Unit 66 The Inscribed Circle and the Right Triangle |
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277 | (5) |
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Unit 67 The Golden Rectangle |
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282 | (5) |
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Unit 68 The Golden Triangle |
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287 | (5) |
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Unit 69 Geometric Fallacies |
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292 | (5) |
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Unit 70 Regular Polyhedra |
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297 | (4) |
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Unit 71 An Introduction to Topology |
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301 | (4) |
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Unit 72 Angles on a Clock |
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305 | (3) |
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Unit 73 Averaging Rates -- the Harmonic Mean |
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308 | (4) |
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312 | (4) |
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Unit 75 Digit Problems Revisited |
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316 | (3) |
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Unit 76 Algebraic Identities |
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319 | (4) |
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Unit 77 A Method for Factoring Trinomials of the Form ax2 + bx + c |
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323 | (3) |
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Unit 78 Solving Quadratic Equations |
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326 | (5) |
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Unit 79 The Euclidean Algorithm |
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331 | (4) |
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335 | (4) |
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Unit 81 Algebraic Fallacies |
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339 | (5) |
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Unit 82 Sum Derivations with Arrays |
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344 | (5) |
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Unit 83 Pythagorean Triples |
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349 | (5) |
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354 | (5) |
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Unit 85 Fibonacci Sequence |
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359 | (5) |
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Unit 86 Diophantine Equations |
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364 | (3) |
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Unit 87 Continued Fractions and Diophantine Equations |
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367 | (5) |
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Unit 88 Simplifying Expressions Involving Infinity |
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372 | (5) |
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Unit 89 Continued Fraction Expansion of Irrational Numbers |
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377 | (5) |
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Unit 90 The Farey Sequence |
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382 | (3) |
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Unit 91 The Parabolic Envelope |
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385 | (5) |
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Unit 92 Application of Congruence to Divisibility |
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390 | (5) |
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Unit 93 Problem Solving -- a Reverse Strategy |
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395 | (6) |
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Unit 94 Decimals and Fractions in Other Bases |
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401 | (2) |
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Unit 95 Polygonal Numbers |
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403 | (8) |
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411 | (5) |
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Unit 97 Angle Trisection -- Possible or Impossible? |
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416 | (3) |
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419 | (5) |
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424 | (4) |
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Unit 100 The Multinomial Theorem |
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428 | (4) |
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Unit 101 Algebraic Solution of Cubic Equations |
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432 | (5) |
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Unit 102 Solving Cubic Equations |
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437 | (7) |
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Unit 103 Calculating Sums of Finite Series |
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444 | (5) |
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Unit 104 A General Formula for the Sum of Series of the Form Σn t=1 tr |
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449 | (6) |
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Unit 105 A Parabolic Calculator |
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455 | (4) |
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Unit 106 Constructing Ellipses |
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459 | (5) |
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Unit 107 Constructing the Parabola |
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464 | (5) |
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Unit 108 Using Higher Plane Curves to Trisect an Angle |
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469 | (4) |
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Unit 109 Constructing Hypocycloid and Epicycloid Circular Envelopes |
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473 | (4) |
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Unit 110 The Harmonic Sequence |
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477 | (3) |
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Unit 111 Transformations and Matrices |
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480 | (6) |
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Unit 112 The Method of Differences |
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486 | (4) |
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Unit 113 Probability Applied to Baseball |
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490 | (3) |
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Unit 114 Introduction to Geometric Transformations |
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493 | (6) |
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Unit 115 The Circle and the Cardioid |
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499 | (5) |
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Unit 116 Complex-Number Applications |
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504 | (5) |
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Unit 117 Hindu Arithmetic |
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509 | (4) |
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Unit 118 Proving Numbers Irrational |
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513 | (3) |
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Unit 119 How to Use a Computer Spreadsheet to Generate Solutions to Certain Mathematics Problems |
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516 | (3) |
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Unit 120 The Three Worlds of Geometry |
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519 | (7) |
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526 | (2) |
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Unit 122 Graphical Iteration |
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528 | (3) |
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Unit 123 The Feigenbaum Plot |
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531 | (3) |
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Unit 124 The Sierpinski Triangle |
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534 | (3) |
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537 | |
