Foolproof, and Other Mathematical Meditations by Brian Hayes
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hen Brian Hayes, a widely respected writer for American Scientist, makes great play of being a nonmathematician, it bothers and surprises me. A mathematician (as far as I’m concerned) is someone who enjoys doing mathematics, qualifications be damned. If Foolproof is anything to go by, Hayes is at least as much of a mathematician as I am. Perhaps being outside of the ivory tower gives Hayes different filters about what is and isn’t worth pursuing. It certainly gives him an angle of approach that makes for fascinating reading; one of my yardsticks for the quality of a pop-mathematics book is ‘‘did it teach me something new?’’ In Foolproof, the answer was yes for almost every one of its thirteen essays. Hayes begins by looking at the well-worn story of the young Carl Friedrich Gauss supposedly getting one over on his teacher, Herr Bu¨ttner, by summing an arithmetic sequence in no time at all. It’s a story I’ve told and embellished many times, giving poor Bu¨ttner a hangover, giving poor Gauss awful numbers to work with, and giving no attention to the truth of the anecdote. The first essay looks at the evolution and genealogy of the account, documenting which of the various details developed where. There’s a surprisingly interesting breakdown of three possible explanations of Gauss’s trick, and how n2 ðn þ 1Þ,
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nðnþ1Þ 2 ,
Þ and n ðnþ1 are all subtly different—and, of course, a 2 note that the trick is not a Gauss original (it goes back to at least Archimedes, and it’s likely that Bu¨ttner would have been aware of the trick himself). In the second essay, ‘‘Beyond the Law of Averages,’’ Hayes defines a distribution he calls the factoidal: using notation I would have avoided, he lets f ?ðnÞ be the number you get if you roll an n-sided die until you get a 1 and take the product of all of the numbers you’ve rolled up to that point. To his surprise, this distribution doesn’t have a well-defined mean: the more samples you take from it, the more often you hit a long run of dice without a 1, leading to an enormous number that dominates everything else in the sample. I especially enjoyed that Hayes documents his puzzlement as he explored this distribution: it is a counterintuitive result, and the story of his developing understanding is what makes the essay charming. (Perhaps this is what he means about being a nonmathematician: what mathematician would be so open about not understanding something? Only the good ones, I’d say.)
‘‘How to Avoid Yourself’’ then looks at the phenomenon of self-avoiding walks, a topic that falls into an interesting area of combinatorics: too numerous to count easily, but too rare for simple approximation methods to find. Hayes takes delight in the richness of the topic and in his own uncertainty—he’s clearly writing his own code to supplement his understanding while simultaneously admiring the researchers at the cutting edge. This essay highlights one of the things I most like about Foolproof: not just that it teaches me things I don’t know, but it asks questions that feel within grasp. Like the
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