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Mathematical Puzzles:A Connoisseur's Collection

Peter Winkler
AK Peters
Publication Date: 
Number of Pages: 
[Reviewed by
James Tanton
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Every six to eight months or so I turn on my office computer and find another e-mail from a colleague or a friend (usually attached with a long header of previous recipients) detailing the latest diabolical mathematical puzzle that's making the electronic rounds. Last year, for instance, the following came my way:

One hundred prisoners and a warden agree to play the following game. Each day, starting tomorrow, the warden will select a prisoner at random and lead her to a room that contains nothing but a lamp standing in the center of the room. The prisoner may do one of three things:

  1. Switch the lamp on if it is off, or switch it off if it is on, and then head back to her cell.

  2. Do nothing and head back to her cell.

  3. Announce: "All 100 prisoners have visited this room."

If a prisoner makes such an announcement, and she is correct, then all 100 prisoners will be set free. If not, all will be executed. The game will continue until someone makes an announcement.

Knowing that they are about to play, what strategy can the 100 prisoners agree upon to ensure their freedom? Assume that the lamp is initially on and that all prisoners know this. (Is there is a winning strategy even if the initial state of the lamp is not known?)

One also hears on the grapevine the latest puzzler big companies are allegedly asking interviewees as a test of their mental faculties. Have you heard this one, for instance?

Downstairs there are three light switches on panel. You are told only that one switch will turn on a light in the attic (which cannot be seen from the basement). With just one trip from the ground floor to the attic is it possible to determine which of the three switches operates the attic light?

(It seems that the topic of light-bulbs was a common theme thrown my way in 2003!) What a delight and relief to read Peter Winkler's wonderful book and find these and a whole host of classic word-of-mouth puzzles posed, solved, and expanded upon, and, when possible, origin revealed. (It seems that the attic light-bulb puzzle is at least a decade old.)

Winkler's book will certainly appeal to the mathematician, as well as to students of all ages — high-school, college, and graduate. His philosophy of what constitutes a good puzzle is right on the mark, showing that this Connoisseur's Collection really is of quality and depth.

A good puzzle, Winkler describes, is easy to state, amusing, and catching, and it should have a sense of "universality" to it — that it is an exploration of a large interesting question rather than a specific curious instance. It should be elegant, both in the ideas presented in its statement and in its solution, and, ideally, have at least one solution that is elementary, insightful, and catches you by surprise. Winkler has certainly collated 120 or so wonderful zingers that fulfill most, if not all, of these requirements. And he doesn't stop when the puzzle is solved, for a good puzzle should also serve as a gateway for further exploration and depth. Winkler next asks about prisoners who face two light bulbs, for instance, in a room without any chance of conferring on a plan of action in advance. He pushes the reader to think about alternative solutions and to follow sources of reference.

He's got the classics — two monks walking up a mountain, the three way duel, fuel needs for a drive around an island, and breaking a chocolate bar, for instance — as well as the not-so-classic and the soon-to-be classic (the infamous hat problem that leads to the discovery of Hamming codes, for example.) I was delighted to also see a chapter of notorious puzzles that remain unsolved today. Winkler includes, with apology, a wonderful chapter on geographical trivia. Although completely non-mathematical, these puzzles serve as great tidbits to throw out to a class when a mathematical lecture begins to lull. (Which US state is closest to the continent of Africa? Answer: Maine!)

The book is beautifully presented on high-quality paper with good formatting. Each chapter is short and succinct, and complete solutions are provided (except the chapter on unsolved puzzles, of course!) Winkler also gives us the heads-up as we go along when generalizations of a puzzle will be explored in a later section of the book. In the role of a mathematical reviewer I give this inspiring book my highest recommendation. (As a mathematician, I have a few more puzzles that Winkler might like to include in his next edition of the text — the banana-carrying camel puzzle, the 100 pirates and 10 coins puzzle, the "making the same potato chip from two different potatoes" puzzle. I also have one question for him: On page 51, does the ham-sandwich theorem really follow as a consequence of just the intermediate value theorem?)

James Tanton, once a professor of mathematics in the college scene, has turned his attention to the challenge and goal of bringing meaningful mathematical thinking to the realm of secondary education. To practice what he preaches, James is now a high-school mathematics teacher (still publishing articles and books in mathematics) working to educate young students and, more importantly, teachers in what mathematics is really about. He offers workshops and professional training courses for all professionals in the field. He can be found at St. Mark's School in Southborough, Massachusetts.

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