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Table of Contents

Acknowledgments xi Introduction 1 Chapter 1: It's Common Knowledge 3 Chapter 2: Simpson's Paradox 11 Chapter 3: The Impossible Problem 21 Chapter 4: Braess's Paradox 31 Chapter 5: The Power of Complex Numbers 39 Chapter 6: Bucking the Odds 50 Chapter 7: Cantor's Paradise 68 Chapter 8: Gamow-Stern Elevators 82 Chapter 9: The Toss of a Coin 88 Chapter 10: Wild-Card Poker 103 Chapter 11: Two Series 113 Chapter 12: Two Card Tricks 131 Chapter 13: The Spin of a Needle 146 Chapter 14: The Best Choice 165 Chapter 15: The Power of Powers 176 Chapter 16: Benford's Law 190 Chapter 17: Goodstein Sequences 201 Chapter 18: The Banach-Tarski Paradox 210 The Motifs 217 Appendix 221 Index 233

About the Author

Julian Havil is a retired former master at Winchester College, England, where he taught mathematics for thirty-three years. In addition to "Nonplussed!", he is the author of "Gamma: Exploring Euler's Constant" (both Princeton).


"Havil once again explores a variety of mathematical results and problems that at first appear to be self-contradictory, or stated in such a way that no solution could exist. In each case, he then either sketches a proof of why the result is not contradictory, or explains the solution to the seemingly unsolvable problem ... Like a magician revealing secrets, Havil maintains this sense through most chapters, dropping the punch line at just the right moment."--J.T. Noonan, Choice "This sequel to the author's book Nonplussed! supplies another set of brain-stretching problems and ideas. Its subtitle is 'Surprising Solutions to Counterintuitive Conundrums'; the surprise often consisting of the fact that it is possible to obtain a solution at all! ... This is another excellent book by Havil, following in the Martin Gardner tradition."--Alan Stevens, Mathematics Today "Julian Havil has quietly joined the ranks of the very best writers of popular mathematics. His two volume set Impossible? and Nonplussed! Mathematical Proof of Implausible Ideas not only belong in every library, but in the hands of every young person interested in mathematics and especially in the hands of their teachers."--John J. Watkins, Mathematical Intelligencer "Impossible? is an immensely thought-provoking book. Even if you skim or skip the more complex abstract math, you may have a hard time letting these puzzles go, so strongly do they flout common sense. You'll just have to do your best to put them our of your mind when you need to get some sleep, but if the situation ever arises, be sure to take Monty up on his offer."--Ray Bert, Civil Engineering "I would highly recommend this book as a reference for the mathematician who likes recreational mathematics, or as a good read for the recreational enthusiast with a penchant for more rigor."--Blair Madore, MAA Reviews

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