*Foolproof* is a collection of essays on mathematical themes. If the snippets above have piqued your interest, you might like the rest of the book too.

The essays explore mathematical ideas that provoke puzzlement and wonder: sets of numbers that have no average, higher-dimensional spaces where spheres shrink away to nothing, curves that are so twisty they cover every point on a plane, a gambling game that you can never win or lose or tie. A few of the essays look toward the research frontier, reporting on topics such as the distribution of prime numbers. Others are historical: A famous anecdote has young Carl Friedrich Gauss outfoxing his math teacher; did it really happen that way? The title essay explores the beautiful but sometimes treacherous idea of mathematical proof.

All of the chapters derive from essays that first appeared as “Computing Science” columns in *American Scientist* magazine; they have been corrected, revised, and updated for this volume. (Links to the original magazine versions are given in the table of contents below.)

*Foolproof, and Other Mathematical Meditations,* by Brian Hayes. The MIT Press. September 2017.

Hardcover edition: ISBN 978-0-262-03686-3, $24.95.

Paperbound edition: ISBN 978-0-262-53607-3, $17.95.

eBook edition: ISBN 978-0-262-34267-4, $17.95.

Ask at your local bookstore.

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Chapter 1. Young Gauss Sums It Up

A schoolboy’s triumph over a harsh teacher becomes the stuff of legend

Chapter 2. Outside the Law of Averages

Not just above average but beyond the reach of all averages

Chapter 3. How to Avoid Yourself

Taking a walk without crossing your own path leads into curious byways

Chapter 4. The Spectrum of Riemannium

An unexpected connection between nuclear physics and prime numbers

Chapter 5. Unwed Numbers

In Sudoku “No math is required,” but there’s plenty behind the scenes

Chapter 6. Crinkly Curves

A scribbled line grows so twisty that it touches every point on a plane

Chapter 7. Wagering with Zeno

You can’t win, you can’t lose, you can’t tie, and the game never ends

Chapter 8. The Higher Arithmetic

New tools for doing arithmetic with enormous numbers

Chapter 9. First Links in the Markov Chain

A crucial innovation in probability began with an analysis of poetry

Chapter 10. Playing Ball in the *n*th Dimension

Spheres in a space of many dimensions seem to shrink away to nothing

Chapter 11. Quasirandom Ramblings

Somewhere between the chaotic and the orderly lies the quasirandom

Chapter 12. Pencil, Paper, and Pi

Forensic mathematics investigates a heroic computation that went awry

Chapter 13. Foolproof

Is proof a magic wand that works only in the hands of wizards?

Hello, I’m Brian Hayes. I write about science, mathematics, computation, and technology. In the 1970s and 80s I was an editor at *Scientific American*, and since 1990 I have been associated with *American Scientist*. The essays in *Foolproof* began life as columns in the latter magazine. I’ve also written and illustrated two other books, *Infrastructure: A Guide to the Industrial Landscape* (W. W. Norton, 2005, 2014) and *Group Theory in the Bedroom, and Other Mathematical Diversions* (Hill and Wang, 2008). My blog is called Bit-Player.

I am not a mathematician, just a writer who became enchanted with mathematical thinking. To put it another way, I have been an avid student of mathematics for more than 50 years, but I still have a lot to learn.

I live in Massachusetts with my wife Rosalind Reid, who is also a science writer. I recently spent a semester in Berkeley, California, as Journalist in Residence at the Simons Institute for the Theory of Computing. I am proud to acknowledge a fellowship from Y Combinator Research, which provided support during the preparation of this book.

If you have questions or comments about *Foolproof*, please send me an email: brian@bit-player.org.

“With a journalist’s instinct for story, a mathematician’s concern for accuracy, and a storryteller’s sense of narrative, Brian Hayes lets the general reader in on a secret mathematicians already know: math is fun! His vignettes are like the snapshots of a returned traveler, showing us exotic lands and the marvelous creatures that live there. *Foolproof* shows that the mathematical enterprise is one of high adventure.”

James Propp, Professor, University of Massachusetts Lowell

“Brian Hayes takes us with him as he roams far and wide across the mathematical landscape. Whether he’s braving the borderlands of the latest research or poking around in some forgotten corner of history, his chronicles of what he finds there are consistently captivating and revelatory.”

Steven Strogatz, Jacob Gould Sherman Professor of Applied Mathematics, Cornell University; author of *The Joy of x*

“Each of these essays brings unexpected twists of perception and presentation; what a fine imagination Hayes has! I enjoyed the book enormously.”

Nick Trefethen FRS, Professor of Numerical Analysis, University of Oxford; creator of Chebfun; author of *Trefethen’s Index Cards*

Dianne Timblin. Adventures in a Less Fretful Cosmos: A Conversation with Brian Hayes. *American Scientist*, September–October 2017, volume 105, pages 312–314. Online. PDF.

“Shecky Riemann”. Overview... *Foolproof, and Other Mathematical Meditations*. *MathTango*. October 8, 2017. Online. See also Books... Year-end Review. *MathTango*. November 24, 2017. Online.

Adhemar Bultheel. Review of *Foolproof, and Other Mathematical Meditations*. European Mathematical Society. October 27, 2017. Online.

Mark Hunacek. Review of *Foolproof, and Other Mathematical Meditations*. Mathematical Association of America. November 11, 2017. Online.

Brian Clegg. Review of *Foolproof, and Other Mathematical Meditations*. *Popular Science: The popular science and science fiction book review site.* November 19, 2017. Online.

Brian Clegg. Brian Hayes—Four-Way Interview. *Popular Science: The popular science and science fiction book review site.* November 26, 2017. Online.

Anna Haensch. Review of *Foolproof, and Other Mathematical Meditations*. *Math Horizons* 25(4):29, March, 2018. Online ($$). Another online source.

William Denton. Review of *Foolproof, and Other Mathematical Meditations*. Miskatonic University Press, 25 April 2018. Online.

Francesca Arici. Review (in Italian) of *Foolproof, and Other Mathematical Meditations*. Math Is in the Air, 16 September 2018. Online.

page 120, four lines from the bottom: “as it might see” should be “as it might seem”

page 182: In describing the errors that William Shanks made in calculating hundreds of digits of *π*, I made some mistake of my own, all too similar to his. Near the top of page 182, the number labeled *U*_{497} is incorrect. The line should read:

*U*_{497} 0.000 ... 9074446680084838973843

This error was discovered by Haruyuki Kawabe, who is preparing a Japanese translation of *Foolproof*.

page 183: Another error pointed out by Haruyuki Kawabe. A diagram on page 183 shows how Shanks may have omitted five digits, probably while transcribing a value from one worksheet to another. Unfortunately, I marked the wrong five digits for deletion. Here is a corrected version of the diagram.