When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible

This book was terrific fun to read! I thought I would skim the chapters to write my review, but I was hooked by the preface, and read through the first 100 pages in one sitting (dinner was late that night). Nahin covers many of the same topics as V.M. Tikhomirov does in Stories About Maxima and Minima (volume 1 in the AMS's Mathematical World Series). In his last story, Tikhomirov says "I think that extremum problems provide wonderful material for teaching thinking, inventiveness, scientific flexibility, and the overcoming of intellectual difficulties (p. 183)," and the book under review overflows with just such material. While the two books are similar in many ways — both embed the mathematics within a historical narrative that motivates the problems and both authors employ a conversational style that talks directly to the reader — Nahin appeals to wider audience and includes discrete optimization problems and methods (probability, linear programming, graph theory) not discussed by Tikhomirov. And the emphasis is different: When Least Is Best is a history of mathematics book with lots of interesting mathematics in it. Stories About Maxima and Minima is a mathematics book with lots of interesting history in it.

So, whom is this book pitched at? Anyone who has taken calculus and a bit of physics should find it accessible. The author is an electrical engineer and makes it clear in the introduction that his themes are "plausibility and/or direct computation (p. 1),"...constructive demonstrations rather than theoretical proofs. I have read quite a few popular science and mathematics books that claim that "the first year of undergraduate calculus and physics is pretty much enough (p. xiii)" as far as prerequisite intellectual background necessary to understand the text. But I often find myself wondering who these gifted undergraduates may be! Nahin does, in fact, adhere to this promise, and although many of the calculations are lengthy and involved, comprehension does require nothing more than patience, perseverance, and first year calculus and physics.

Here are the features of this book that make it so engaging. First is the author's obvious delight and enjoyment — he is having fun and it is contagious. He is particularly enamored of paradoxical results. For example if we toss four fair dice, the most likely number of 3's to show is zero. But an even more likely event is that at least one 3 shows. "This strikes many as a paradoxical result, but that is part of the inexhaustible charm of probability! (p. 9)." Within each chapter there are teasers that egg the reader on by stating problems, facts or flaws that he will justify, prove or explain later. Within the flow of a demonstration he will state a result or generalization and give a tantalizing clue, then invite the reader to see if they can figure it out before he does so later in the chapter. Certain problems are woven throughout the book (Tikhomirov also does this) solved in different ways, connecting the mathematical ideas and encouraging the reader to look at a situation from various angles. Finally, there is no bibliography and ordinarily this would not be listed as a positive feature. However, relevant sources are included as part of the text. I appreciated not having to flip to a list of references at the back of the book, or even move my eyes to the bottom of the page to check a footnote. The citation became part of the story, not interrupting the flow. And the references are eclectic and wide ranging, including Thomas Heath's A History of Greek Mathematics, William Dunham's Journey Through Genius, Herman Melville's Moby Dick, Rebecca Goldstein's Strange Attractors, articles from Isis, Mathematics Teacher, The College Mathematics Journal, Mathematics Magazine, The American Mathematical Monthly, American Journal of Physics, Scientific American, and many more.

So exactly who is the audience for this book? It would make an excellent gift for a bright high school student who has taken or is taking AP Calculus, or her college counterpart. I've already decided to buy a few copies to offer as prizes for a student poster session, and give to my favorite graduating seniors. Mathematics professors could assign individual chapters or parts of chapters as projects in calculus, differential equations, modeling, and perhaps some other classes. I can imagine using the book as a text for a seminar, independent study, or thesis. The references could lead to advanced explorations. I can also imagine generating a series of math club talks (again, following through on some of the references as well) led by students. And, in case I haven't made it perfectly clear, every mathematically inclined colleague, relative or friend will thoroughly enjoy this delicious book.

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Bonnie Shulman (bshulman@abacus.bates.edu) is associate professor of mathematics at Bates College in Lewiston, ME and is looking for clever ways to make the time it takes to grade large stacks of papers as small as possible. Her current research is in history and philosophy of mathematics and she is having a wonderful time teaching a senior capstone seminar on History of the Proof, tracing the development of Abel's proof of the insolvability of the quintic by radicals.