Customer Reviews

Nonplussed!: Mathematical Proof of Implausible Ideas

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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.

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.

I read Impossibles first and really enjoyed it a lot. This was also enjoyable, but I found myself skimming over the proofs much of the time. I did not do that with Impossibles (but I don't remember there being as much). The problems discussed were ineresting, but I did not find myself telling my other geek friends about very many.

I have read a number of books of this type over the years. Some are a model of clarity and are truly a pleasure to read. At the other end of the spectrum, there are those that are seriously lacking in this regard and are a chore to go through. In the case of this book, I would place it closer to the latter end of the range. Consequently, I would agree with the prior reviewers who have expressed similar sentiments. More specifically, I often found myself bewildered at how some of the initial equations were set up since very little in the way of explanation was given. Much more space was devoted to going through calculations line by line - something which, in my view, is of much less importance. On the positive side, the author has addressed more than a dozen problems of different types, thus potentially appealing to readers with different interests. I my case, I found a few of these problems to be quite fascinating while in the case of several others, I had no interest whatsoever - the lack of clarity causing me to skim over the material, as in the case of a prior reviewer. Finally, I do agree with the author that the mathematics level for most of the problems is up to about the senior high school level; however, in a few cases, the mathematics is well beyond that.

Overall, I was rather disappointed with this book. Although clearly not ideal for the general reader, this book may appeal to serious math buffs with a very broad range of interests and perhaps a refined intuition to enhance their understanding of what the author is trying to convey.

Another one of these books written by someone who feels that he does not have to express himself clearly, just "cleverly." I, for one, resent having to figure out what an author is trying to say--clarity is everything!!!!. Mr. Havil needs to be informed that the rigor demanded in a textbook is not the same as that in a volume for the public at large.

From the first chapter on the two coin problem, I knew I was in trouble when I could not figure out from the exposition what the correct answer is. After being put off by the notation on the tennis problems which I could not understand and which the author did not disdain to explain and daunted by the complex mathematics involved (despite the author's assurance that some high school mathematics is all that is needed) I gave up.

This type of conceited garbage writing about one of the greatest discplines is what has given mathematics the bad name it has. Perhaps, Mr. Havil should have beta-tested the book rather than rushing inconsiderately into print.

In short, I looked foward to an entertaining and informative discussion of counterintuitiveness and I ended up regretting my purchase of the booik and my knowledge of its existence.

Robert Allen

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.