Princeton Puzzlers

John J. Watkins is professor emeritus of mathematics at Colorado College. An award-winning teacher, he is the author of Topics in Commutative Ring Theory (Princeton) and coauthor of Graphs: An Introductory Approach. Across the Board is the definitive work on chessboard problems. It is not simply about chess but the chessboard itself--that simple grid of squares so common to games around the world. And, more importantly, the fascinating mathematics behind it. From the Knight's Tour Problem and Queens Domination to their many variations, John Watkins surveys all the well-known problems in this surprisingly fertile area of recreational mathematics. Can a knight follow a path that covers every square once, ending on the starting square? How many queens are needed so that every square is targeted or occupied by one of the queens? Each main topic is treated in depth from its historical conception through to its status today. Many beautiful solutions have emerged for basic chessboard problems since mathematicians first began working on them in earnest over three centuries ago, but such problems, including those involving polyominoes, have now been extended to three-dimensional chessboards and even chessboards on unusual surfaces such as toruses (the equivalent of playing chess on a doughnut) and cylinders. Using the highly visual language of graph theory, Watkins gently guides the reader to the forefront of current research in mathematics. By solving some of the many exercises sprinkled throughout, the reader can share fully in the excitement of discovery. Showing that chess puzzles are the starting point for important mathematical ideas that have resonated for centuries, Across the Board will captivate students and instructors, mathematicians, chess enthusiasts, and puzzle devotees. "This book is extremely well written and is, no doubt, the best exposition of the connection between the chessboard problems and recreational mathematics. The author surveys all the well-known problems about chess and the chessboard. . . . The problems are treated in depth from their beginnings through to their status today."---Mohammed Aassila, MAA Review "Torus-shaped boards, three-dimensional boards, a shape called the Klein bottle--the simple checkerboard pattern proves to be creatively malleable when Watkins puts his mind to his hobbylike subject. Watkins' invitational tone ensures attention from the finite but enthusiastic audience for mathematical recreation." "Watkins offers an excellent invitation to serious mathematics." "I would be happy to recommend this book to you. . . . The book is an easy and entertaining read that shows numerous paths into various branches of discrete mathematics and graph theory."---Paul J. Campbell, Mathematics Magazine "This is not just about chess, but also the three centuries of 'recreational mathematics' that the game has inspired. From simple questions, such as whether it is possible for a knight to land on each square of the board on its path, Watkins wades into graph theory, the mathematics of three-dimensional chess and even chess on a torus." "This book is stimulating and very well written. It is admirably clear. . . . Definitely the book is highly recommended and is of much interest. This book is, no doubt, the newly best exposition of the interconnection between amusing recreational mathematics and the interesting chessboard problems. I feel sure that it will be of great use both to students of graph theory, geometry, topology and mathematics, in general, and captivate to scholars, instructors, chess enthusiasts, puzzle devotees, and to those intervening in amusing and recreational mathematics."---Francisco José Cano Sevilla, European Mathematical Society "A most enjoyable book that will surely offer new and original avenues for problem solvers of all kinds in need of new techniques, approaches or problems to solve."---Robert Bilinski, Crux "This beautiful book is absolutely the best treatmen

Robert B. Banks (1922-2002) was the author of Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics (Princeton). He was professor of engineering at Northwestern University and dean of engineering at the University of Illinois at Chicago. Have you ever daydreamed about digging a hole to the other side of the world? Robert Banks not only entertains such ideas but, better yet, he supplies the mathematical know-how to turn fantasies into problem-solving adventures. In this sequel to the popular Towing Icebergs, Falling Dominoes (Princeton, 1998), Banks presents another collection of puzzles for readers interested in sharpening their thinking and mathematical skills. The problems range from the wondrous to the eminently practical. In one chapter, the author helps us determine the total number of people who have lived on earth; in another, he shows how an understanding of mathematical curves can help a thrifty lover, armed with construction paper and scissors, keep expenses down on Valentine's Day. In twenty-six chapters, Banks chooses topics that are fairly easy to analyze using relatively simple mathematics. The phenomena he describes are ones that we encounter in our daily lives or can visualize without much trouble. For example, how do you get the most pizza slices with the least number of cuts? To go from point A to point B in a downpour of rain, should you walk slowly, jog moderately, or run as fast as possible to get least wet? What is the length of the seam on a baseball? If all the ice in the world melted, what would happen to Florida, the Mississippi River, and Niagara Falls? Why do snowflakes have six sides? Covering a broad range of fields, from geography and environmental studies to map- and flag-making, Banks uses basic algebra and geometry to solve problems. If famous scientists have also pondered these questions, the author shares the historical details with the reader. Designed to entertain and to stimulate thinking, this book can be read for sheer personal enjoyment. "[Banks displays] a playful imagination and love of the fantastic that one would not ordinarily associate with a mathematical engineer. . . . Banks's style is entertaining but never condescending." "Banks turns trivial questions into mind-expanding demonstrations of the magical powers of mathematics. Nor does he restrict himself to trivial questions: his shrewd analyses coax secrets out of such weighty topics as global population growth and the melting of polar ice caps. . . . Not a math textbook which teaches readers how to solve set types of problems, this collection of puzzles does something far more important: it teaches us how to delight in unexpected challenges to our numerical imagination."

"One of Choice's Outstanding Academic Titles for 1999" Robert B. Banks (1922-2002) was Professor of Engineering at Northwestern University and Dean of Engineering at the University of Illinois at Chicago. He served with the Ford Foundation in Mexico City and with the Asian Institute of Technology in Bangkok. He won numerous national and international honors, including being named Commander of the Order of the White Elephant by the King of Thailand and Commandeur dans l'Ordre des Palmes Academiques by the government of France. He is the author of Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics (Princeton). Although we seldom think of it, our lives are played out in a world of numbers. Such common activities as throwing baseballs, skipping rope, growing flowers, playing football, measuring savings accounts, and many others are inherently mathematical. So are more speculative problems that are simply fun to ponder in themselves--such as the best way to score Olympic events. Here Robert Banks presents a wide range of musings, both practical and entertaining, that have intrigued him and others: How tall can one grow? Why do we get stuck in traffic? Which football player would have a better chance of breaking away--a small, speedy wide receiver or a huge, slow linebacker? Can California water shortages be alleviated by towing icebergs from Antarctica? What is the fastest the 100-meter dash will ever be run? The book's twenty-four concise chapters, each centered on a real-world phenomenon, are presented in an informal and engaging manner. Banks shows how math and simple reasoning together may produce elegant models that explain everything from the federal debt to the proper technique for ski-jumping. This book, which requires of its readers only a basic understanding of high school or college math, is for anyone fascinated by the workings of mathematics in our everyday lives, as well as its applications to what may be imagined. All will be rewarded with a myriad of interesting problems and the know-how to solve them. Some images inside the book are unavailable due to digital copyright restrictions. "Robert Banks's study of everyday phenomena is infused with infectious enthusiasm." "There is something here for every mathematically inclined reader. The aerodynamics of balls in sport, the spread of diseases, traffic flow, the effect of meteor impacts--[Banks] deals with these and much more in engaging, well-judged detail."---Robert Matthews, New Scientist "A fabulous exposition of adventures in applied mathematics. It's already one of my favourite books. It's so good I find it hard to lay aside."---B. L. Henry, Physicist "This book provides an entertaining look at some simple and interesting mathematical models for a range of topics. . . . The choice of modeling subjects is imaginative. . . . Every chapter is interesting, and the self-contained nature of each section of the book means that one can happily 'dip in and out' without losing the thread of the text."---Alistair Fitt, London Mathematical Society Newsletter "The book stands out because the examples are all treated as real-life examples with real data, and taking into account all the complications that are usually left out in academic examples: the earth is not a perfect sphere, a baseball is rough because of its stitches, it is thrown with spin, there is resistance of the air, and the resistance differs with the height, etc. Even though, there are a lot of formulas and numbers, the reading is pleasant and smooth."---A. Bultheel, European Mathematical Society

Paul J. Nahin is the best-selling author of many popular math books, including Mrs. Perkins's Electric Quilt, Digital Dice, Dr. Euler's Fabulous Formula, When Least Is Best, and An Imaginary Tale (all Princeton). He is professor emeritus of electrical engineering at the University of New Hampshire. We all played tag when we were kids. What most of us don't realize is that this simple chase game is in fact an application of pursuit theory, and that the same principles of games like tag, dodgeball, and hide-and-seek are also at play in military strategy, high-seas chases by the Coast Guard, and even romantic pursuits. In Chases and Escapes, Paul Nahin gives us the first complete history of this fascinating area of mathematics, from its classical analytical beginnings to the present day. Drawing on game theory, geometry, linear algebra, target-tracking algorithms, and much more, Nahin also offers an array of challenging puzzles with their historical background and broader applications. Chases and Escapes includes solutions to all problems and provides computer programs that readers can use for their own cutting-edge analysis. Now with a gripping new preface on how the Enola Gay escaped the shock wave from the atomic bomb dropped on Hiroshima, this book will appeal to anyone interested in the mathematics that underlie pursuit and evasion. Some images inside the book are unavailable due to digital copyright restrictions. "In the 18th century, mathematicians began to tease apart how best to track down and intercept prey, inspired by pirate ships bearing down on merchant vessels. The mathematics is by no means trivial, and quickly becomes fiendish if the merchant ship takes evasive action. This is just one of the colorful problems in Paul Nahin's fascinating history of the mathematics of pursuit, in which he guides us masterfully through the maths itself--think lions and Christians, submarines and torpedoes, and the curvaceous flight of fighter aircraft." "This is a highly readable book that offers several colorful applications of differential equations and good examples of non-trivial integrals for calculus students. It would be a good source of examples for the classroom and or a starting point for an independent project."---Bill Satzer, MAA Review "This book contains a well-written, well-organized collection of solutions to twenty-one challenging calculus and differential equation problems that concern pursuit and evasion as well as the historical background of each problem type." "I am sure that this book will appeal to everyone who is interested in mathematics and game theory. Excellent work."---Prabhat Kumar Mahanti, Zentralblatt Math "Chases and Escapes is a wonderful collection of interesting and classic pursuit and evasion problems. . . . If you are interested in in dogs chasing ducks, pirates chasing merchants, and submarines hiding, then this book is for you." "Nahin provides beautiful applications of calculus, differential equations, and game theory. If you are pursuing an enjoyable collection of mathematical problems and the stories behind them, then your search ends here."-Arthur Benjamin, Harvey Mudd College "I know of no better way to grasp the basic concepts of calculus than to study pursuit-and-escape problems. Paul Nahin has made a superb survey of the vast field of such problems, from Zeno's paradox of Achilles and the tortoise through the famous four bugs that once made the cover of Scientific American. Not only does he make clear the required differential equations, but he traces each problem's colorful history. No book on the topic could be more definitive or a greater pleasure to read."-Martin Gardner "Chases and Escapes is a superb treatment of the solutions to a variety of pursuit-evasion problems, some classic and others more contemporary. The content is accessible to undergraduates in mathematics or the physical sciences, with lots of supporting detail included. The author's lively wr

Paul J. Nahin is professor emeritus of electrical engineering at the University of New Hampshire. He is the author of many best-selling popular math books, including Digital Dice, Chases and Escapes, Dr. Euler's Fabulous Formula, When Least Is Best, Duelling Idiots and Other Probability Puzzlers, and An Imaginary Tale (all Princeton). Fun puzzles that use physics to explore the wonders of everyday life Physics can explain many of the things that we commonly encounter. It can tell us why the night is dark, what causes the tides, and even how best to catch a baseball. With In Praise of Simple Physics, popular math and science writer Paul Nahin presents a plethora of situations that explore the science and math behind the wonders of everyday life. Roaming through a diverse range of puzzles, he illustrates how physics shows us ways to wring more energy from renewable sources, to measure the gravity in our car garages, to figure out which of three light switches in the basement controls the light bulb in the attic, and much, much more. How fast can you travel from London to Paris? How do scientists calculate the energy of an atomic bomb explosion? How do you kick a football so it stays in the air and goes a long way downfield? Nahin begins with simpler problems and progresses to more challenging questions, and his entertaining, accessible, and scientifically and mathematically informed explanations are all punctuated by his trademark humor. Readers are presumed to have some background in beginning differential and integral calculus. Whether you simply have a personal interest in physics' influence in the world or you're an engineering and science student who wants to gain more physics know-how, this book has an intriguing scenario for you. In Praise of Simple Physics proves that if we look carefully at the world around us, physics has answers for the most astonishing day-to-day occurrences. "The energy of moving water, the physics of communication satellites and the maths behind catching a ball are all skilfully dissected by engineer and writer Paul Nahin in this enjoyable study of everyday physics." "Nahin's writing style, as in previous books, is clear, conversational, humorous and chatty. . . . [A]nd the discussions in the book are careful and appropriately rigorous." "Fun, accessible physics/math problems along with some humor."---Antonio Cangiano, Math-Blog "[Nahin] knows how to catch the attention of his reader. You will not regret buying any of his books, and I am sure after reading it, you will pick up this one to check again on one of his models and his solution methods." "A superb book. . . . [D]emonstrates clever ways to solve simply physics problems." "Praise for the previous edition: "Fun, accessible physics [and] math problems along with some humor."" "In Praise of Simple Physics offers a masterful look at how basic principles, combined with clever thinking and fundamental mathematics, lead to satisfying explanations of an extraordinary range of natural phenomena, from the path of a football to why the sky is dark at night. Nahin's humor and accessible examples shine. Celebrate your curiosity and delve into this magnificent book!"-Paul Halpern, author of Einstein's Dice and Schrödinger's Cat "In Praise of Simple Physics is a must-read for anyone curious about how physics shows up in everyday life. Paul Nahin's clear explanations demystify the mathematics behind the physics, and with over two dozen applications, his book is a great introduction to mathematical modeling as well."-Oscar Fernandez, author of Everyday Calculus "This is a superb book-thoughtful, historical, carefully done, and intensely clever. It was a joy to read."-Christopher G. Tully, Princeton University "Many books that explore classical physics problems tend to be relentlessly sequential and completely humorless. In this book, Paul Nahin shows a lighter touch and he gives himself the necessary space to tell the mathematical story. T

Paul J. Nahin is the author of many best-selling popular-math books, including Chases and Escapes, Dr. Euler's Fabulous Formula, When Least is Best, Duelling Idiots and Other Probability Puzzlers, and An Imaginary Tale (all Princeton). He is professor emeritus of electrical engineering at the University of New Hampshire. Some probability problems are so difficult that they stump the smartest mathematicians. But even the hardest of these problems can often be solved with a computer and a Monte Carlo simulation, in which a random-number generator simulates a physical process, such as a million rolls of a pair of dice. This is what Digital Dice is all about: how to get numerical answers to difficult probability problems without having to solve complicated mathematical equations. Popular-math writer Paul Nahin challenges readers to solve twenty-one difficult but fun problems, from determining the odds of coin-flipping games to figuring out the behavior of elevators. Problems build from relatively easy (deciding whether a dishwasher who breaks most of the dishes at a restaurant during a given week is clumsy or just the victim of randomness) to the very difficult (tackling branching processes of the kind that had to be solved by Manhattan Project mathematician Stanislaw Ulam). In his characteristic style, Nahin brings the problems to life with interesting and odd historical anecdotes. Readers learn, for example, not just how to determine the optimal stopping point in any selection process but that astronomer Johannes Kepler selected his second wife by interviewing eleven women. The book shows readers how to write elementary computer codes using any common programming language, and provides solutions and line-by-line walk-throughs of a MATLAB code for each problem. Digital Dice will appeal to anyone who enjoys popular math or computer science. In a new preface, Nahin wittily addresses some of the responses he received to the first edition. "The problems are accessible but still realistic enough to be engaging, and the solutions in the back of the book will get you through any sticky spots. Writing your own versions of a few of these programs will acquaint you with a useful approach to problem solving and a novel style of thinking."---Brian Hayes, American Scientist "Digital Dice will appeal to recreational mathematicians who have even a limited knowledge of computer programming, and even nonprogrammers will find most of the problems entertaining to ponder." "[An] enjoyable read, as [Nahin] writes clearly, with humour and is not afraid to include equations where necessary. Nahin spices the book throughout with factual and anecdotal snippets. Digital Dice will appeal to all who like recreational mathematics."---Alan Stevens, Mathematics Today "[T]he book is targeted at teachers and students of probability theory or computer science, as well as aficionados of recreational mathematics, but anyone who is familiar with the basics of probability and is capable of writing simple computer programs will have no problem working their way through this interesting and rewarding book." "After the appearance of the author's earlier book on probability problems, [Duelling Idiots And Other Probability Puzzlers], one has high expectations for this book, and one is not disappointed. . . . The book will certainly have great appeal to all three of the targeted audiences."---G A. Hewer, Mathematical Reviews "This well-written entertaining collection of twenty-one probability problems presents their origin and history as well as their computer solutions. . . . These problems could be used in a computer programming course or a probability course that includes Monte Carlo simulations."---Thomas Sonnabend, Mathematics Teacher "All of the books by Nahin and Havil are worth having, including others not listed here. I particularly recommend Digital Dice for the task of teaching undergraduates in mathematics the fundamentals of computation a

Paul J. Nahin is the best-selling author of many popular math books, including Mrs. Perkins's Electric Quilt, Digital Dice, Dr. Euler's Fabulous Formula, When Least Is Best, and An Imaginary Tale (all Princeton). He is professor emeritus of electrical engineering at the University of New Hampshire. What are your chances of dying on your next flight, being called for jury duty, or winning the lottery? We all encounter probability problems in our everyday lives. In this collection of twenty-one puzzles, Paul Nahin challenges us to think creatively about the laws of probability as they apply in playful, sometimes deceptive, ways to a fascinating array of speculative situations. Games of Russian roulette, problems involving the accumulation of insects on flypaper, and strategies for determining the odds of the underdog winning the World Series all reveal intriguing dimensions to the workings of probability. Over the years, Nahin, a veteran writer and teacher of the subject, has collected these and other favorite puzzles designed to instruct and entertain math enthusiasts of all backgrounds. If idiots A and B alternately take aim at each other with a six-shot revolver containing one bullet, what is the probability idiot A will win? What are the chances it will snow on your birthday in any given year? How can researchers use coin flipping and the laws of probability to obtain honest answers to embarrassing survey questions? The solutions are presented here in detail, and many contain a profound element of surprise. And some puzzles are beautiful illustrations of basic mathematical concepts: "The Blind Spider and the Fly," for example, is a clever variation of a "random walk" problem, and "Duelling Idiots" and "The Underdog and the World Series" are straightforward introductions to binomial distributions. Written in an informal way and containing a plethora of interesting historical material, Duelling Idiots is ideal for those who are fascinated by mathematics and the role it plays in everyday life and in our imaginations. "Nahin's sophisticated puzzles, and their accompanying explanations, have a far better than even chance of fascinating and preoccupying the mathematically literate readership they seek." "An entertaining, thought-provoking collection of twenty-one puzzles. . . .These puzzles invite the reader to think intuitively, mathematically, and creatively about the laws of probability as they apply in lighthearted, often counterintuitive ways to a diverse collection of practical and speculative situations." "By following Nahin's informal style it is possible to set [the examples] up quickly from first principles and slip them into courses on calculus, algebra, or scientific programming. They also offer a wealth of topics for undergraduate projects. Those duelling idiots are fighting over a goldmine."---Des Higham, MSOR Connections "For those of us who thoroughly enjoy a good puzzle, Duelling Idiots is indeed a welcome book. What Paul Nahin offers is essentially the mathematical equivalent of a collection of Far Side cartoons: a series of quirky vignettes, each with an amusing punchline that reveals something new about an offbeat aspect of reality."-Mark Denny, Stanford University "Duelling Idiots and Other Probability Puzzlers seeks to teach the fundamentals of elementary probability theory using topics that are familiar to most everyone. Its light-hearted way of explaining serious subjects is a refreshing approach."-Robert B. Banks, author of Towing Icebergs, Falling Dominoes and of Slicing Pizzas, Racing Turtles