Mathematics is a tool for understanding the world we live in. Mathematics is also a world unto itself, detached from the material universe, a realm built not from matter and energy but from the stuff of imagination. In the mathematical universe we can not only explore but also create, inventing new geometries, new kinds of numbers, even new kinds of logic. Mathematics, in other words, can take you out of this world. Bertrand Russell, who charted the territory of mathematics a century ago, spoke of opening “windows into a larger and less fretful cosmos.”

Crinkly Curves

In 1877 the German mathematician Georg Cantor made a shocking discovery. He found that a two-dimensional surface contains no more points than a one-dimensional line. Intuition rebels against this notion. Inside a square you could draw infinitely many parallel line segments side by side. Surely an area with room for such an infinite array of lines must include more points than a single line—but it doesn’t. Cantor himself was incredulous: “I see it, but I don’t believe it,” he wrote.

Quasirandom Ramblings

Suppose you want to estimate the area of an object with a complicated shape, such as a maple leaf. There’s a well-known trick for solving this problem with the help of a little randomness. Put the leaf on a board of known area, then throw darts at it randomly, trying not to aim. If a total of N darts hit the board, and n of them land within the leaf, then the ratio n:N approximates the ratio of the leaf area to the board area.

How to Avoid Yourself

Every Sunday morning you go for a walk in the city, heading nowhere in particular, with just one rule to your rambling: You never retrace your steps or cross your own path. If you have already walked along a certain block or passed through an intersection, you refuse to set foot there again. This recipe for tracing a loopless path through a grid of city streets leads into some surprising back alleys of mathematics. Avoiding yourself, it turns out, is a hard problem.

First Links in the Markov Chain

It was an unlikely partnership of poetry and probability theory. Delving into the text of Alexander Pushkin’s novel-in-verse Eugene Onegin, the Russian mathematician A. A. Markov spent hours sifting through patterns of vowels and consonants. His analysis did not alter the understanding or appreciation of Pushkin’s poem, but the technique he developed—now known as a Markov chain—extended the theory of probability in a new direction. And it gave us a marvelous tool for turning literature into drivel.


I was a teenage angle trisector. In my first full-time job, fresh out of high school, I trisected angles for $1.75 an hour. My employer was a maker of voltmeters, ammeters, and other electrical instruments. This was in the analog age, when a meter had a slender pointer swinging in an arc across a scale. My job was drawing the scale, using ruler and compass and a fine pen. Often I would have to divide an angle into three equal parts—a trisection. I joked about this with my coworker. We should get extra pay, I said, for solving one of the most famous unsolvable problems of antiquity.

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.)

Publication details

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

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

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

Where to get it

Ask at your local bookstore.

Order from the publisher.

Order from Amazon.

See if a nearby library has a copy.


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 nth 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?

About the author

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. In 2018 I’ll be 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 of comments about Foolproof, please send me an email:

Nice words from my friends

“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

Reviews and Commentary

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.

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

Updates and corrections

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