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Nonplussed! Mathematical Proof of Implausible Ideas

Julian Havil
Princeton University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
P. N. Ruane
, on

A title like this might suggest theoretical discussion on matters of plausibility and proof (perhaps conducted within the context of some psychological model of mathematical understanding, such as that given by Skemp [1 ]). And the outcome of such discourse could be of use to anyone teaching mathematics from the infant stages and upwards. After all, isn’t it important for teachers to be able to distinguish between ideas that are intrinsically difficult, as opposed to those that can be made plausible? The long division algorithm, for instance, usually remains a mystery — even to those who can use it; and a statement such as 1= 0.999… will be regarded as being totally implausible by many good high school students. Then there is the matter of distinguishing between highly implausible facts (e.g. dense sets may have measure zero) and those that are not intuitively obvious (e.g. Pythagoras’ Theorem).

However, in no way to its detriment, Julian Havil’s book isn’t concerned with such generalities. Instead, its fourteen short chapters examine a range of problems yielding outcomes strongly at variance with intuitive expectations. About half of the book is concerned with situations of a probabilistic nature, whilst the remainder deal with various themes such as the mechanical paradox of Leyburn’s Uphill Roller, Toricelli’s Trumpet, Pursuit Problems and a chapter on Hyperdimensions (e.g. surface area in hyperdimensions). The treatment, however, isn’t solely mathematical, since the problems are discussed from a variety of general perspectives (historical, biographical and practical relevance). The most salient feature of the book is the elegance of its presentation and the lucidity of Havil’s exposition. This, together with the appealing range of illustrations, mean that an initial scan of the book’s 200 pages creates an instant feeling of ‘must read’.

As for probability, there are chapters on familiar themes, such as Buffon’s needle and the counterintuitive outcome concerning the likelihood of any two persons within a group having the same birthday. But Havil always reveals surprising aspects of problems like this. For instance, the chapter on the ‘Birthday Paradox’ generalises the situation and looks at applications to coding and cryptography. The later chapter, ‘Toss of a Needle’, also considers variations on Buffon’s problem, but alongside many interesting historical observations. Subsequently, Chapter 13 uses statistical methods to explain why the 13th day of the month falls more frequently on a Friday than any other day of the month. Less familiar probabilistic themes include a chapter called ‘Parrondo's Games’, which reveals the apparently paradoxical situation whereby individually losing strategies can be combined to form a winning one. But the very first chapter considers similar ideas relating to the vagaries of the scoring system in tennis and team selection.

In terms of readership, this book can be read by (or used with) anyone with a good background in high school mathematics onwards. The mathematics of each problem is explained in such a way as to consolidate prior knowledge, and the use of less familiar techniques is always well motivated. For example, in the chapter called ‘Toricelli’s Trumpet’, calculus is elegantly employed to determine the surface area and volume of a solid of revolution, preceded by historical allusions to Thomas Hobbes, John Wallis, Cavalieri and, of course, Toricelli himself. Later in the book, there is the lovely chapter on ‘Hyperdimensions’, which considers volumes in discrete and continuous hyperdimensions. In the process, it provides applications for integral reduction formulae and establishes the relevance of gamma functions to the maximisation of volumes of hyperspheres.

Finally, although this book is based upon a group of seemingly disparate mathematical problems, the consistency of the author’s narrative style, and thematic continuity of the text, means that there is no hint of fragmentation. It is therefore recommended that Julian Havil’s headmaster award him further sabbatical leave for the purpose of producing a sequel to this welcome addition to the mathematical literature.


[1] The Psychology of Learning Mathematics, R.R. Skemp, Penguin Books, 1972

Peter Ruane is retired from a rewarding (but non-lucrative) career in the training of primary and secondary mathematics teachers.

 Preface xi
Acknowledgements xiii
Introduction 1
Chapter 1: Three Tennis Paradoxes 4
Chapter 2: The Uphill Roller 16
Chapter 3: The Birthday Paradox 25
Chapter 4: The Spin of a Table 37
Chapter 5: Derangements 46
Chapter 6: Conway's Chequerboard Army 62
Chapter 7: The Toss of a Needle 68
Chapter 8: Torricelli's Trumpet 82
Chapter 9: Nontransitive Effects 92
Chapter 10: A Pursuit Problem 105
Chapter 11: Parrondo's Games 115
Chapter 12: Hyperdimensions 127
Chapter 13: Friday the 13th 151
Chapter 14: Fractran 162
The Motifs 180
Appendix A: The Inclusion-Exclusion Principle 187
Appendix B: The Binomial Inversion Formula 189
Appendix C: Surface Area and Arc Length 193
Index 195