There is, of course, no shortage of books that attempt to explain interesting mathematical ideas to laypeople, but over the years I have found that they tend to be of variable quality. Some are so watered-down that they really only convey a false sense of understanding, others are so technical that they can’t really be understood by the intended audience, and others are, on occasion, just plain wrong. It takes some skill to write a popular book that is accurate, accessible and genuinely informative. Brian Hayes has that skill, and *Foolproof *is one of those books.

Though I have never read any other books by Hayes, the name was nonetheless somewhat familiar to me, and I soon realized why: over the years I have reviewed four books in the *Best Writing on Mathematics* annual anthology series, and three of those four books (2012, 2014 and 2016) contain articles by him. (In fact, the articles in the 2012 and 2014 issues appear as chapters here, as does an entry by Hayes in the 2010 anthology, which I have not seen.) It is not surprising that Hayes’ work shows up in Best Writing compilations, as he definitely has a way with words, as we can see even without venturing beyond the Preface:

I am not a mathematician — not a native citizen of the Republic of Numbers. But I have been living there, an expatriate litterateur, for most of my adult years. I have struggled to learn the language, immersed myself in the culture and customs, and become an enthusiastic amateur practitioner. My life has been greatly enriched by the experience.

This text consists of 13 essays, each of which appeared (in some form, anyway; they have been updated and in some cases expanded for this volume) in *American Scientist* magazine.

Considerations of space make it impracticable to describe in detail each of the chapters that comprise this book, but I can describe a few of them.

The first chapter is actually more historical than mathematical; in it, Hayes delves deeply into the famous story about how Gauss, given a busywork assignment by a teacher to add a large number of consecutive numbers, added them almost instantly with no calculation. As I learned this story many years ago, the numbers were the first 100 positive integers, and Gauss added them by rewriting the sum S = (1 + … + 100) as (100 + ….+ 1), adding the two together to get (100)(101), and then dividing by 2. This is not the way Hayes first learned the story, however; he has Gauss doing the sum in a different way. That inconsistency is at the heart of this article: Hayes carefully examines the historical evidence and concludes that the story may have grown in the telling, and that there is no uniformity on what numbers Gauss was asked to add or just how he did it. He also considers the issue of who, before Gauss, knew how to do this.

The fifth chapter is on the mathematics behind the game of Sudoku. This is the subject of an entire book (*Taking Sudoku Seriously* by Rosenhouse and Taalman), but here Hayes provides a nice survey of some of the basic questions discussed in that book: How many completed Sudoku puzzles are there? How many “essentially different” puzzles are there? What is the minimum number of clues that are necessary to ensure that a puzzle always has a unique solution? (One quibble: the Rosenhouse and Taalman book doesn’t appear in the bibliography for this article. It definitely should.)

The final chapter — the one that gives this book its title — is a discussion about the nature of proof. This chapter is a beautiful rumination on the role of proof in mathematics, addressing such diverse topics as proofs that are too long to read, proofs that use computers, and proofs that prove something is impossible (e.g., the inability to trisect an arbitrary angle with compass and straightedge). For a person who is, by his own account, “not a native citizen of the Republic of Numbers”, Hayes demonstrates a very impressive breadth of knowledge about rather advanced mathematics (e.g., the *abc *conjecture), and this article is one that I think all undergraduate math majors could profit from reading.

There is, of course, a great deal more. Other articles, for example, discuss the history of space-filling curves, interesting stuff that can happen in *n*-dimensional space, random walks, and the decimal expansion of \(\pi\). (This last article is another one that should interest historians. It discusses William Shanks, who, over a long period of time in the last half of the 19th century, computed \(\pi\) to 707 decimal places — but made a mistake at around decimal place 527 that affected all the remaining digits. By diligent computer work, Hayes attempts to discover just what this mistake could have been.)

The book ends with a chapter-by-chapter bibliography. I was amused to see that the author was aware of certain aspects of his sources, such as the fact that Bell’s *Men of Mathematics* could not be uncriticially relied on as a source of factual information: “Bell has a reputation as a highly inventive writer (a trait not always considered considered a virtue in a biographer or historian).”

This compilation is a treasure trove of high-quality mathematical exposition. I saw nothing in it that a reader would later have to “unlearn” as being false, and although the chapters were sufficiently demanding that they would not insult the intelligence of a reader, they were accessible enough so that a layperson would likely get something out of them. For a sense of what the chapters are like, there is an online reprint of an article in *American Scientist* that became chapter 13 of the book now under review. (I plan on making this link available to the students in my courses this semester on geometry and the history of mathematics.) All of the chapters in this book, in fact, originally appeared in preliminary form in *American Scientist*, and finding PDF versions of these original articles online is not hard; in fact, the author generously provides them on his blog, “bit player”.

But don’t let the availability of these articles deter you from buying the book itself — the chapters in the book are generally improved and expanded versions of the original articles, and it is worth the relatively small price of the book to have them available in one convenient source. This is a book that will convey to any reader the elegance and beauty of mathematics. Undergraduate students should certainly be made aware of its existence, and faculty members, who already know that mathematics is elegant and beautiful, can use this book as fodder for course material or lectures to a math club. All told, this is a useful and enjoyable book, and is highly recommended.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.