| BUT FIRST, A WORD FROM OUR SPONSORS |
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| WELCOME! |
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| SURFING THE BOOK |
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| CHAPTER ONE Fun and Games |
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2 | (36) |
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An Introduction to Rigorous Thought |
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1.1 Silly Stories Each with a Moral |
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4 | (10) |
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Conundrums that Evoke Techniques of Effective Thinking |
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14 | (4) |
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Leading Questions and Hints for Resolving the Stories |
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18 | (9) |
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Solutions and Further Commentary |
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27 | (11) |
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Discovering Strategies of Thought for Life |
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| CHAPTER TWO Number Contemplation |
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38 | (98) |
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40 | (9) |
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How the Pigeonhole Principle Leads to Precision Through Estimation |
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2.2 Numerical Patterns in Nature |
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49 | (15) |
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Discovering the Beauty of the Fibonacci Numbers |
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2.3 Prime Cuts of Numbers |
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64 | (18) |
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How the Prime Numbers Are the Building Blocks of All Natural Numbers |
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2.4 Crazy Clocks and Checking Out Bars |
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82 | (13) |
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Cyclical Clock Arithmetic and Bar Codes |
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2.5 Public Secret Codes and How to Become a Spy |
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95 | (15) |
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Encrypting Information Using Modular Arithmetic and Primes |
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2.6 The Irrational Side of Numbers |
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110 | (11) |
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Are There Numbers Beyond Fractions? |
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121 | (15) |
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The Point of Decimals and Pinpointing Numbers on the Real Line |
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| CHAPTER THREE Infinity |
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136 | (70) |
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138 | (7) |
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3.2 Comparing the Infinite |
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145 | (17) |
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Pairing Up Collections via a One-to-One Correspondence |
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162 | (11) |
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Georg Cantor Answers: Are Some Infinities Larger Than Others? |
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3.4 Travels Toward the Stratosphere of Infinities |
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173 | (17) |
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The Power Set and the Question of an Infinite Galaxy of Infinities |
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3.5 Straightening Up the Circle |
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190 | (16) |
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Exploring the Infinite Within Geometrical Objects |
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| CHAPTER FOUR Geometric Gems |
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206 | (120) |
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4.1 Pythagoras and His Hypotenuse |
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208 | (10) |
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How a Puzzle Leads to the Proof of One of the Gems of Mathematics |
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4.2 A View of an Art Gallery |
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218 | (14) |
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Using Computational Geometry to Place Security Cameras in Museums |
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4.3 The Sexiest Rectangle |
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232 | (17) |
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Finding Aesthetics in Life, Art, and Math Through the Golden Rectangle |
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4.4 Smoothing Symmetry and Spinning Pinwheels |
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249 | (20) |
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Can a Floor Be Tiled Without Any Repeating Pattern? |
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4.5 The Platonic Solids Turn Amorous |
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269 | (20) |
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Discovering the Symmetry and Interconnections Among the Platonic Solids |
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4.6 The Shape of Reality? |
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289 | (18) |
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How Straight Lines Can Bend in Non-Euclidean Geometries |
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307 | (19) |
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| CHAPTER FIVE Contortions of Space |
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326 | (76) |
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5.1 Rubber Sheet Geometry |
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328 | (18) |
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Discovering the Topological Idea of Equivalence by Distortion |
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5.2 The Band That Wouldn't Stop Playing |
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346 | (13) |
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Experimenting with the Mobius Band and Klein Bottle |
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359 | (15) |
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Exploring Relationships Among Vertices, Edges, and Faces |
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374 | (15) |
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Untangling Ropes and Rings |
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5.5 Fixed Points, Hot Loops, and Rainy Days |
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389 | (13) |
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How the Certainty of Fixed Points Implies Certain Weather Phenomena |
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| CHAPTER SIX Chaos and Fractals |
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402 | (112) |
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404 | (8) |
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Viewing a Gallery of Fractals |
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6.2 The Dynamics of Change |
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412 | (18) |
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Can Change Be Modeled by Repeated Applications of Simple Processes? |
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6.3 The Infinitely Detailed Beauty of Fractals |
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430 | (28) |
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How to Create Works of Infinite Intricacy Through Repeated Processes |
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6.4 The Mysterious Art of Imaginary Fractals |
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458 | (24) |
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Creating Julia and Mandelbrot Sets by Stepping Out in the Complex Plane |
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482 | (21) |
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How Repeated Simple Processes Result in Utter Chaos |
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503 | (11) |
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Can the Dimensions of Fractals Fall Through the Cracks? |
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| CHAPTER SEVEN Taming Uncertainty |
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514 | (114) |
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516 | (7) |
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Some Scenarios Involving Chance That Confound Our Intuition |
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7.2 Predicting the Future in an Uncertain World |
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523 | (18) |
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How to Measure Uncertainty Using the Idea of Probability |
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541 | (13) |
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Are Coincidences as Truly Amazing as They First Appear? |
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554 | (17) |
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Systematically Counting All Possible Outcomes |
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7.5 Collecting Data Rather than Dust |
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571 | (14) |
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The Power and Pitfalls of Statistics |
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7.6 What the Average American Has |
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585 | (25) |
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Different Means of Describing Data |
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7.7 Parenting Peas, Twins, and Hypotheses |
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610 | (18) |
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Making Inferences from Data |
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| CHAPTER EIGHT Deciding Wisely |
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Applications of Rigorous Thinking |
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628 | (2) |
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630 | (15) |
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Deciding How to Weigh the Unknown Future |
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645 | (18) |
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Deciding Personal and Public Policy |
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663 | (19) |
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Deciding Between Faring Well and Welfare |
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682 | (18) |
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Deciding Who Actually Wins an Election |
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8.5 Cutting Cake for Greedy People |
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700 | (16) |
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Deciding How to Slice Up Scarce Resources |
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| FAREWELL |
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716 | (3) |
| ACKNOWLEDGMENTS: SECOND EDITION |
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719 | (3) |
| ACKNOWLEDGMENTS: FIRST EDITION |
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722 | (3) |
| HINTS AND SOLUTIONS |
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725 | (22) |
| INDEX |
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747 | (11) |
| CREDITS |
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758 | |
Other versions by this Author
