Wednesday Book Club: Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi

Wednesday, 24 December 2008 — 11:26pm | Book Club, Literature, Mathematics

This week’s selection: Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi (2008) by Martin Gardner.

In brief: This revised anthology of Martin Gardner’s “Mathematical Games” columns in Scientific American, the first of fifteen volumes, is an ample exhibition of the author’s repute as the canonical journalist of recreational mathematics. Though the brevity of the articles leaves the details of proofs bottled up in the extensive bibliography, the non-technical approach goes a long way towards illustrating the everyday relevance of esoterica in topology and combinatorial theory.

(The Wednesday Book Club is an ongoing initiative of mine to write a book review every week. I invite you to peruse the index. For more on Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi, keep reading below.)

I first became aware of Martin Gardner through his successor at Scientific American, Douglas Hofstadter, whose vignettes on self-referential sentences, Chopin, Rubik’s Cube, the Lisp programming language, and a whole host of other Things I Like, appeared in Metamagical Themas: Questing for the Essence of Mind and Pattern. Hofstadter’s reverence for Gardner, who wrote the “Mathematical Games” columns at the magazine from 1956 to 1981, was more than enough to keep my eye open for Gardner’s work. To be frank, I am surprised that I hadn’t discovered it earlier: Gardner is considered an authority on a number of subjects that are near and dear to me, among them pseudoscience and Lewis Carroll.

Needless to say, I was rather delighted to see that the Cambridge University Press is issuing new editions of Gardner’s fifteen volumes of articles under the heading of The New Martin Gardner Mathematical Library, with an all-star editorial board that includes names like John Conway and Donald Knuth. I pulled the first two volumes off the shelf on sight, and judging by Hexaflexagons, I intend to collect them all.

Hexaflexagons consists of sixteen articles originally published from 1956 to 1958, with annotations and postscripta that have accumulated over several reprintings. The topics are a broad sample of what Gardner refers to as “recreational mathematics”—that is, mathematics as it is applied to the world of card tricks, logic puzzles, and board games.

Most of the articles accord with one or both of two rhetorical formulae. The first is the explication of unexpected correspondences between one problem and another, like the familiar Tower of Hanoi and the traversal of the nodes on a flattened topological map of a polyhedron. The second is the generalization of small, contained problems into larger problem domains, as in the illustrations of how to play tic-tac-toe on a four-dimensional hypercube, or how polyominoes—interlocking permutations of n square tiles that should be familiar to anyone who has played Tetris—can or cannot tesselate rectangular grids of varying sizes and shapes.

Many of the problems in the book will be familiar to modern readers with an interest in puzzles, but even so, there is some historical curiosity to be shared in realizing that Gardner’s articles were often the first to introduce them to the general public. Gardner provides solutions for a few representative problems in each article, as well as two columns that consist entirely of unrelated puzzles, but leaves much as an exercise for the reader (or at least, the scrupulous reader who does not stoop to looking up the solutions on Google).

As the articles are directed to a general audience, most of the underlying mathematics is explained in words and diagrams, and the few equations that appear never exceed the bounds of high-school algebra. Mathematical literacy is still an asset, however, for the sake of working out the myriad proofs that Gardner casually mentions on the byway. If there is one frustration to be had with the book, it is Gardner’s understandable habit of dangling the tantalizing existence of a proof in front of the reader, but leaving the explanations to the sources in each article’s comprehensive bibliography. This is an acceptable price to pay for the elucidation of advanced concepts to the public without intimidating them, and perhaps it is to Gardner’s credit that he leaves the audience wanting more. If his intent is to foster interest in mathematics, I would call the execution a success.

Gardner is at his best when he seizes diversions that are putatively trivial and expands them into a broader context. The chapter on “probability paradoxes”, or counterintuitive results in probability theory, is a fine example. It begins with a few choice illustrations—why you are more likely to be holding two aces if you declare not “I have an ace,” but “I have the ace of spades”; why the chance of a pair of a shared birthdays in a group of 24 individuals is slightly better than half—but quickly segues to a treatment of one of the most important questions in the philosophy of science: what examples we consider relevant as the confirming instances of a hypothesis. To cite Gardner’s case study: in theory, does the discovery of a purple cow not improve the likelihood of the statement, “All crows are black”? Or, to cite Gardner’s parody of Gelett Burgess:

I never saw a purple cow,
But if I ever see one,
Will the probability crows are black
Have a better chance to be 1?

What we can say with certainty is this: Gardner’s crystalline writing is both a lucid presentation of evergreen curiosities and an elegant introduction to the theories behind their analysis. Problems that were fresh in the 1950s may look ordinary today, but it takes a keen interpreter to evoke their underlying intricacy in plain English. It should be clear to any reader who picks up Hexaflexagons that Gardner’s work is of monumental importance not because he was the first on the scene, but because he knew how to communicate that most elusive of messages: why something is fun.

Here, that “something” is mathematics. That is a good thing; why, I shall leave as an exercise to the reader.


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2 rejoinders to “Wednesday Book Club: Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi

  1. Good Lord. For someone with your interests not to have known from infancy, almost, who Martin Gardner is and why he’s important is like discovering an American (as Hofstadter says in another book) who doesn’t know where Chicago is.

    Congratulations on updating yourself, and much joy to you as you make your way through GardnerWorld. It’s a wonderful place.

    Friday, 26 December 2008 at 11:34am

  2. Yes, that is exactly how I felt.

    Friday, 26 December 2008 at 12:04pm

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