Nonplussed!: Mathematical Proof of Implausible Ideas | Julian Havil

Nonplussed!: Mathematical Proof of Implausible Ideas | Julian Havil | A real brain teaser

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Nonplussed!: Mathe...

	Nonplussed!: Mathematical Proof of Implausible Ideas Julian Havil Princeton University Press, 2007 - 208 pages average customer review:based on 3 reviews view larger image
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Math--the application of reasonable logic to reasonable assumptions--usually produces reasonable results. But sometimes math generates astonishing paradoxes--conclusions that seem completely unreasonable or just plain impossible but that are nevertheless demonstrably true: Conclusions that, for example, tell us that a losing sports team can become a winning one by adding worse players than its opponents. Or that the thirteenth of the month is more likely to be a Friday than any other day. Or that cones can roll unaided uphill. In Nonplussed!--a delightfully eclectic collection of paradoxes from many different areas of math--popular-math writer Julian Havil reveals the math that shows the truth of these and many other unbelievable ideas.

Nonplussed! pays special attention to problems from probability and statistics, areas where intuition can easily be wrong. These problems include the vagaries of tennis scoring, what can be deduced from tossing a needle, and disadvantageous games that form winning combinations. Other chapters address everything from the historically important Torricelli's Trumpet to the mind-warping implications of objects that live on high dimensions. Readers learn about the colorful history and people associated with many of these problems in addition to their mathematical proofs.

Nonplussed! will appeal to anyone with a calculus background who enjoys popular math books or puzzles.

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Fascinating

This book will delight readers who like to get their hands into their math. Havil sticks to mostly elementary concepts, avoiding highly abstract fields that would lose most readers. When a subject could go too far afield, Havil warns about it and presents only the part the reader needs to know, citing original source references for the interested reader. He gives complete, understandable proofs of some startling statements--proofs that leave you understanding exactly how you got there. The great thing is that you can choose to work through these problems for yourself, verifying each step, or you can just follow along with his proofs and take on faith any simple algebraic rearrangements that he may have skipped over. Compared to Havil's earlier classic on Euler's Gamma Function, this one's a bit easier to read, with numerous short sections on a variety of topics.

One minor complaint is that I found some typesetting errors. One, ironically, occurs on page 49 where he uses the notation "!n" (the number of derangements of n objects) when actually he meant "n!" (the number of permutations of n objects). It's ironic because only two paragraphs later Havil warns that !n can be easily confused with n!, whereupon he adopts a new notation for !n. In the delightfully bizarre but challenging chapter on John Conway's Fractran, there are a few typos that might confuse that minority of readers who will actually try to go line-by-line through the explanation of the Fractran machine (p. 172), but if you're one of those people, discovering the errors will anyway prove your mastery.

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A real brain teaser

The book of Julian Havil is certainly not easy reading. Perhaps I am a dummy, but at several pages I had to read over a paragraph several times before understanding its real meaning, but the result was always worth the trouble. The calculations itself are explained thoroughly and his way of highlighting different sidesteps are often eye-openers.
People loving Martin Gardner's articles in Scientific American, will certainly appreciate this book.

Mathematically Impeccable--Real World Flawed

This book is a valuable addition to a math-puzzler's library, but contains some flaws on real-world data.

For example, Havil shows, with impeccable mathematics, that if a given player has over 91.9643...% probability of winning any given point on his or her serve, that he or she has a higher likelihood of winning at the start of the game than when the score is 30-15 or 40-30. He uses this fact to back up a claim that "a high quality tennis player serving at 40-30 or 30-15 to an equal opponent has less chance of winning the game than at its start." Again, this is predicated on that 92% or better percentage of winning any given point. But in real life, high quality tennis players, even when serving, against an equal opponent does not have this high a percentage of the points gained. Take 92% as the percentage. That would mean that over 70% of the time, the non-server would not even get one point (score of 15) during a given game. If anyone watches Wimbledon or the U.S. Open, one sees that such occurrences are rare, not common. As even Havil points out, it also implies that the server will win at least 99.9% of the games. But in high-level play, set scores of 6-3, 6-4, etc. are common. With 99.9% of the games being won by the server, 99.4% of sets would go into tie-break. That's clearly not the case in the real world. But this discrepancy is needed in order to make the "paradox" that creates the "nonplussed" reaction.

In the chapter on the calendar, Havil explains why the Christian feast commemorating Jesus' ascension into Heaven never falls on a Sunday by claiming that that feast is also called Holy Thursday. It's not. It's Ascension Thursday. Holy Thursday, 42 days (six weeks) before Ascension Thursday, is the day before Good Friday, and commemorates the Last Supper.

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