Group Theory in the Bedroom, and Other Mathematical Diversions

"Brian Hayes, since 1993 the 'Computing Science' columnist for American Scientist magazine, is an unrepentant numbers nut. His second book (following 2005's Infrastructure: A Field Guide to the Industrial Landscape) complies and updates approximately a decade's worth of his most significant articles, creating an elegant and high-minded overview of how rigorous mathematical laws intersect with and govern our daily lives."—Mike Newirth, Time Out Chicago



"He's a graceful writer. And if you love numbers, grids and graphs, you'll love this book."—Sara Lippincott, Los Angeles Times Book Review

"Stimulating . . . Brian Hayes discusses how he got caught up in figuring out how to precisely locate the Continental Divide. He began pondering the issue in Idaho during a road trip, and, as he puts it, he was unusually compelled to work it out in his head: 'For a week or so I had no choice but to think about the problem' . . . Hayes isn't a mathematician, which is often to his credit: His best pieces have a journalistic looseness that dovetails nicely with the academic rigor he brings to the subject matter. Indeed, the opening essay, on the immense (and immensely complex) clock at Strasbourg Cathedral, is a masterpiece of science writing. Hayes stands in awe of the clock's capabilities—the 160-plus years-old timepiece has an error rate of 'less than a second per century' and can accurately account for leap years, Easter and other temporal changes. But he's also willing to provide some detail on how the clock literally ticks, and insert some insightful riffing on the notion of clocks that last for millennia. ('To assume that the values of our own age embody eternal verities and virtues is foolish and arrogant,' he writes.) Like any science writer who successfully writes for the layperson, Hayes has an ear for the poetic. Discussing a book on the 'mathematics of armed conflict' by Lewis Fry Richardson, Hayes smartly calls out a lovely two-word sentence about how difficult it is to locate the start- and endpoints of wars: 'Thinginess fails' . . . Though Hayes tries to bring a casual feel to such subjects as partitioning (breaking up numbers into equal subsets) and base-three counting, he's in deep woods. 'You might stumble onto the sequence 0102010, which is square free but cannot be extended without creating a square,' he writes, before enthusing, 'Try it!' Must I? Better this, though, than the many lesser inheritors of The Tipping Point and Freakonomics, writers who figure that writing on science is largely a matter of finding a provocative study and asserting wide-reaching applications from it. While Hayes is an assured and genial guide through the often thorny wilds of computation and mathematics, he never promises more than he can deliver. When discussing the complexities of wealth-distribution models, he claims no fix for economic injustice; there and elsewhere, he's content to simply present the terms of discussion and argue that the numbers and graphs at his command are beautiful, playful things. Quite often, he's right."—Mark Athitakis, The Star Tribune (Minneapolis)



"It is ever so reassuring to discover that scientific, indeed mathematical, scholarship and essay writing is alive and well. Brian Hayes, who has written many columns for The Sciences magazine, here gives his past and future readers a real treat. With a deft hand at explanation and with feet firmly on the ground of practicality, the author is able to take ordinary mundane phenomena and bring them to life mathematically. The first essay points out difficulties in making accurate clocks which predict dates of Easter as well as days, hours, minutes and seconds, not to say sun and moon rising and setting: a great deal lies behind the construction of practical gear trains for such purposes, a theme taken up again later. Subsequent essays contemplate randomness, genetic codes, studies of patterns in the outbreaks of conflict, how to locate the (or a) continental divide, hard problems (NP completeness), problems with choosing names for new objects, efficiency of calculation in various bases, and whether there are many or there is only one electron! The final essay, from which the book takes its title, provides a welcome explanation to the many perplexed purchasers of mattresses that carry instructions to turn them over regularly so as not to develop lumps and hollows without clear indication of how. Highly recommended and an excellent present for anyone who is curious about the world in which they live, and appreciative of how true scholarship goes well beyond information retrieval to make surprising but significant and meaningful connections."—John H. Mason, Mathematical Reviews

"Group Theory in the Bedroom, and Other Mathematical Diversions is a marvelous collection of thought-provoking essays that both inform and entertain. You'll be amazed by the things you'll discover in these stories."—Ron Graham, professor of mathematics, computer science and engineering, University of California, San Diego

"Brian Hayes's book is a refreshing collection of superb mathematical essays. Ranging from choosing up sides to choosing names, the topics are intriguingly nonstandard. Moreover, the writing is clean, the explanations are pellucid, and the effect on the reader is exhilarating. First-rate all the way through."—John Allen Paulos, author of Innumeracy: Mathematical Illiteracy and Its Consequences and Irreligion: A Mathematician Explains Why the Arguments for God Just Don't Add Up

"Every essay in this book is a gem of science writing on its highest level—accurate, up to date, brimming with surprising information, deep insights, and a profound love of mathematics. Its scope is awesome. Topics include a fantastic clock in Strasbourg, randomness, poverty, war, geology, genetics, gear ratios, partitions, nomenclature, group theory, and the ambiguity of the equals sign. There isn't a dull page in the book."—Martin Gardner, author of The Colossal Book of Short Puzzles and Problems



"While driving across my home state of Michigan not so long ago, I noticed that the format for the state's license plates changed. The old format had three letters followed by three digits; sometime recently, a fourth digit was added. Wondering why the change had been made, I figured that the old format had 26³ · 1000 possible designations and quickly estimated that this gave around 16 million possibilities. I knew that the population of Michigan is about 10 million so I could see, assuming license plates are not reused, how we could be running low on plates either now or in the near future. Maybe this was a good time to add another digit. Mathematicians do this kind of thinking frequently. There's no deep mathematics involved, but our comfort with computation can help explain observations we make as well as make a long drive a little less tedious. In his wonderful new book, Group Theory in the Bedroom, and Other Mathematical Diversions, Brian Hayes shows how pleasurable this king of thinking can be and the surprising places to which it can lead. The book collects and slightly reworks twelve of Hayes' essays, which originally appeared over the last decade in the The Sciences and American Scientist. Each essay also includes an 'Afterthoughts' section in which Hayes reflects on the original work, scrupulously own up to errors, and discusses responses received from readers . . . While the other essays do not originate in a long car trip, many of them read like travel essays, as Hayes takes us on expeditions through landscapes surrounding interesting mathematical questions. Sometimes, such as when the continental divide appears before our eyes, we realize the joy has been in the journey as much as the destination. Other times, these investigations lead to surprising questions that reflect more broadly on who we are, what we value, and the nature of mathematics . . . Most of the essays contain interesting bits of what could be called trivia, though they never seem superfluous. There are often interesting biographical sketches of little-known players as well as tidbits such as how random numbers are used in the Ethernet networking protocol; the origin of the equals sign; and why Brocot, a French watchmaker, was led to discover what we now call the Stern-Brocot tree. The writing is uniformly good, exceptionally clear, and with abundant humor and humanity. Hayes writes with a refreshing openness; as part of his colloquial style, he is quick to admit errors and naive ideas. Group Theory in the Bedroom should be accessible to a wide audience. Hayes uses almost no mathematical notation and assumes relatively little mathematical knowledge . . . The essays, taken as a whole, demonstrate convincingly the joy of mathematical thinking and the real power available when it is applied to more general inquiries. As much as any book I can name, Group Theory in the Bedroom conveys to a general audience the playfulness involved in doing mathematics: how questions arise as a form of play, how our first attempts at answering questions usually seem naive in hindsight but are crucial for finding eventual solutions, and how a good solution just feels right . . . In addition, Hayes' writing, with its openness, invites the reader to participate actively. I often felt I was having a conversation, and at times an argument, with the author . . . Without addressing the issue explicitly, Group Theory in the Bedroom presents a more compelling argument for the importance of mathematics. Whether it develops general skills or not, mathematics is a fundamental tool in an intellectual toolkit and is crucial for making sense of the world around us."—David Austin, Notices of the AMS

"A selection of 'Computing Science' columns by American Scientist magazine's former editor-in-chief aimed at the numerate-or at least mathematically curious-reader . . . The first essay explains how clockmakers developed the gears and linkages that enabled fabled medieval clocks to reach remarkable accuracy, as well as predict the day Easter would fall on. Other essays celebrate the notion of random numbers and why they are so hard to achieve. Numerical analysis also plays a role in economic models based on the kinetic theory of gases or simplified markets involving iterations of buying and selling. Hayes goes on to explain how statistics have been applied to compute which quarrels—from interpersonal to world wars—are the deadliest (surprising results here). Also, he looks at how algorithms have been developed to determine ways to divide a random series of numbers into two parts with equal sums, or nearly equal sums if the series total is odd. Gears appear again in the form of algorithms, which yield practical tables of numbers to enable engineers to make gear trains to approximate complex ratios. A couple of essays probe areas only professionals might ponder, such as computing the location of the Continental Divide or why base 3 arithmetic is better than base 10 or binary systems. But the piece de resistance is the title essay, which explains why there is no algorithm whose repetitions would cycle through all four possible mattress positions that would assure equal wear and tear over time. Challenging but rewarding for anyone intrigued by numbers."—Kirkus Reviews

"If your idea of fun includes puzzling over the creation of an algorithm for the Continental Divide, then this essay collection by the former editor in chief of American Scientist will tickle your imagination. Hayes, now an award-winning columnist for American Scientist, has put together some of his best pieces and has included with each a section called 'Afterthoughts,' in which he enthusiastically adds new information and humbly corrects old mistakes. Hayes explores topics as diverse as the centuries-old Strasbourg clock, economic theory, randomness, DNA, gear ratios, weather forecasting, and war and international relations. And with tongue firmly in cheek, he even writes about the ways that one can flip a mattress. Although one need not be a rocket scientist—or even an undergraduate math major—to understand Hayes's work, the wit and elegance of the essays are best appreciated by those with a solid math background and an interest in math play. Recommended for large public libraries and academic libraries supporting programs in mathematics and computer science."—School Library Journal

"In charming prose . . . science and technology journalist Hayes explains the engineering and arithmetic of clocks and gears, wracks his brain over questions of how best to flip a mattress and visits 'the prettiest wrong idea in all of twentieth-century science . . . the vision of piglets suckling on messenger RNA.' As he examines huge calculating tables rendered obsolete by computers, Hayes 'cannot help wondering which of my labors will appear equally quaint and pathetic to some future reader.' This observation is echoed by the afterwords where Hayes addresses pointed questions and observations from readers, displaying a brave willingness to admit error and acknowledge advances made since these pieces were first published in The Sciences and American Scientist. Present-day readers would do best to approach this collection more for its literary merits than its revelation of obscure history or cutting-edge mathematical theory."—Publishers Weekly



Table of Contents
Clock of Ages
Random Resources
Follow the Money
Inventing the Genetic Code
Statistics of Deadly Quarrels
Dividing the Continent
On the Teeth of Wheels
The Easiest Hard Problem
Naming Names
Third Base
Identity Crisis
Group Theory in the Bedroom