# Nonplussed!: Mathematical Proof of Implausible Ideas

Julian Havil
Pages: 216
https://www.jstor.org/stable/j.ctt7t03s

1. Front Matter
(pp. i-viii)
(pp. ix-x)
3. Preface
(pp. xi-xii)
4. Acknowledgements
(pp. xiii-xvi)
5. Introduction
(pp. 1-3)

It does not take a student of mathematics long to discover results which are surprising or clever or both and for which the explanations themselves might enjoy those same virtues. In the author’s case it is probable that in the long past the ‘coin rolling around a coin’ puzzle provided Carroll’s beginning and a welcome, if temporary, release from the dry challenges of elementary algebra:

Two identical coins of equal radius are placed side by side, with one of them fixed. Starting head up and without slipping, rotate one about the other until it is on the other side of...

6. Chapter 1 THREE TENNIS PARADOXES
(pp. 4-15)

In this first chapter we will look at three examples of sport-related counterintuitive phenomena: the first two couched in terms of tennis, the third intrinsically connected with it.

The late Leo Moser posed this first problem during his long association with the University of Alberta. Suppose that there are three members of a club who decide to embark on a private tournament: a new member M, his friend F (who is a better player) and the club’s top player T.

M is encouraged by F and by the offer of a prize if M wins at least two games in...

7. Chapter 2 THE UPHILL ROLLER
(pp. 16-24)

The Proceedings of the Old Baileydated 18 April 1694 chronicles a busy day devoted to handing down justice, in which 29 death sentences were passed as well as numerous orders for brandings; there would have been 30 death sentences had not one lady successfully ‘pleaded her belly’ (that is, proved that she was pregnant). The business part of the document ends with a list of the 29 unfortunates and continues to another list; this time of advertisements (rather strange to the modern mind), which begins with the following paragraph:

THE Ladies Dictionary: Being a pleasant Entertainment for the Fair...

8. Chapter 3 THE BIRTHDAY PARADOX
(pp. 25-36)

Perhaps one of the most well-known examples of a counterintuitive phenomenon concerns the likelihood of two individuals in a gathering sharing the same birthday. If we ignore leap years, then, with a gathering of 366 people, we are assured of at least one repetition of a birthday (a simple application of the subtly powerful Pigeon Hole Principle). That observation is clear enough. What is considerably more perplexing is the size of the group which would result in a 50:50 chance of such a repetition; intuition has commonly argued that, since we halve the probability, we should need about half the...

9. Chapter 4 THE SPIN OF A TABLE
(pp. 37-45)

Martin Gardner brought to the wider world ‘a delightful combinatorial problem of unknown origin’ in his February 1979 column inScientific American. He commented that Robert Tappay of Toronto had passed it to him, who believed it to have originated in Russia:

A square table stands on a central column, which allows it to rotate freely in a horizontal plane. At each corner there is a pocket too deep to allow the contents to be seen and of a size to accommodate an ordinary, empty wine glass. An electronic mechanism is fitted so that, with each pocket containing a single...

10. Chapter 5 DERANGEMENTS
(pp. 46-61)

We shall look at a famous old problem in three different, enlightening ways and then consider three surprising facts originating from it.

The French word for 13,trieze, was also the name of a commonly played card game of the eighteenth century. It could be considered as a simple patience (or solitaire) game but in its classic form it was played by several individuals, and commonly for money. We will leave it to the man who is credited with its first analysis to explain matters:

The players draw first for who will have the hand. We suppose that this is...

11. Chapter 6 CONWAY’S CHEQUERBOARD ARMY
(pp. 62-67)

John Horton Conway is very hard to encapsulate. He is universally acknowledged as a world-class mathematician, a claim strongly substantiated by his occupation of the John von Neumann Chair of Mathematics at Princeton University. His vast ability and remarkable originality have caused him to contribute significantly to group theory, knot theory, number theory, coding theory and game theory (among other things); he is also the inventor of surreal numbers, which seem to be the ultimate extension of the number system and, most famous of all in popular mathematics, he invented the cellular automata game of Life. In chapter 14 we...

12. Chapter 7 THE TOSS OF A NEEDLE
(pp. 68-81)

TheSociety for the Diffusion of Useful Knowledge, founded (mainly by Lord Brougham) in 1828, had the object of publishing information for people who were unable to obtain formal teaching, or who preferred self-education. The celebrated English mathematician and logician Augustus De Morgan was a gifted educator who contributed no less than 712 articles to one of the society’s publications, thePenny Cyclopaedia: one of them (published in 1838 and titledInduction) detailed (possibly for the first time) a rigorous basis for mathematical induction.

It would appear that De Morgan was contacted by more than his fair share of people...

13. Chapter 8 TORRICELLI’S TRUMPET
(pp. 82-91)

One of the longest and most vitriolic intellectual disputes of all time took place between the two seventeenth-century luminaries Thomas Hobbes and John Wallis: Hobbes, the philosopher, had claimed to have ‘squared the circle’ and Wallis, the mathematician, had strongly and publicly refuted that claim.

This ancient problem (one of three of its kind) had been handed down by the Greeks and asked if it was possible, using straight edge and compasses only, to construct a square equal in area to the given circle: it took until 1882 until Ferdinand Lindemann provedπto be transcendental, which meant that the...

14. Chapter 9 NONTRANSITIVE EFFECTS
(pp. 92-104)

A dictionary definition of the adjective ‘transitive’ is ‘being or relating to a relationship with the property that if the relationship holds between a first element and a second and between the second element and a third, it holds between the first and third elements.’

Initially it is easy to imagine that all meaningful relationships between all pairs of objects are transitive: ‘older than’, ‘bigger than’, etc., but we do not need to look too far to produce examples for which transitivity fails: ‘son of’, ‘perpendicular to’, etc. This chapter is primarily concerned with a relationship which is seemingly transitive,...

15. Chapter 10 A PURSUIT PROBLEM
(pp. 105-114)

The defunct magazineGraham DIALwas circulated to 25 000 American engineers during the 1940s and featured aPrivate Corner for Mathematicians, edited by L. A. Graham himself and populated by problems posed by readers for other readers to solve. Akin to Martin Gardner’s articles inScientific American, the articles spawned two books which discussed, commented on and sometimes extended the original contributions. The first book,Ingenious Mathematical Problems and Methods, was published in 1959 and contains the problem we will discuss here. It is framed as a chase on the high seas, and does not seem to have sufficient...

16. Chapter 11 PARRONDO’S GAMES
(pp. 115-126)

We are all used to the idea of losing in any number of games of chance, particularly in games which are biased against us. If we decide to alleviate the monotony of giving our money away by varying play between two such games, we would reasonably expect no surprises in the inevitability of our financial decline, but that is to ignore the discovery of Dr Juan Parrondo:two losing games can combine to a winning composite game.

We will avoid the rigorous definition of a losing game since we all have an instinctive feel for what one is, and that...

17. Chapter 12 HYPERDIMENSIONS
(pp. 127-150)

Some dimensionally dependent phenomena seem reasonable. Take, for instance, the idea of a random walk. In one dimension this means that we start at the origin and move to the left or the right with equal probability; in two dimensions we have four equally probable directions in which we can walk; in both cases it can be shown that the probability of eventually returning to the origin is 1; we cannot, in theory, get lost. As the dimension increases, so we might reasonably think are the chances of getting lost, never to return to the origin, increase, and so they...

18. Chapter 13 FRIDAY THE 13TH
(pp. 151-161)

The bottom right-hand slot of the letters page of the LondonTimesis often reserved for offbeat or amusing correspondence and was occupied on Friday, 13 February 1970, by the following:

Sir,

If, as some of your recent correspondents suggest, eccentricity is one of the criteria for publication of letters to theTimes, you may be willing to allow me, on this doubly unlucky date of Friday the Thirteenth of February, to remind any superstitious among your readers that the 13th day of the month falls more frequently on a Friday than upon any other day of the week.

This...

19. Chapter 14 FRACTRAN
(pp. 162-179)

In chapter 6 we looked at one idea from the fertile and original mind of John Conway, and now we will look at a second, which, in typical whimsical style, he called ‘fourteen fantastic fractions’ in his joint publication with Richard Guy,The Book of Numbers. The idea appeared earlier, in his articleFractran: A Simple Universal Programming Language for Arithmetic, which constituted chapter 2 of the 1987 bookOpen Problems in Communication and Computation(ed. T. M. Cover and B. Gopinath), Springer, pp. 4–26. Further articles about the construction abound; we have used one of Richard Guy’s from...

20. The Motifs
(pp. 180-186)

Southwest Africa finds its most celebrated place on the mathematical map on the border of Uganda and Zaire, since it was there in 1960 that the Belgian geologist Jean de Heinzelin discovered on the shores of Lake Edward the ancient Ishango Bone; its provenance is disputed with its date varying from 8000 to 20 000 B.C.E. and its purpose from a lunar calendar to a list of prime numbers. Yet there are other African ethnomathematical treasures, and the attractive designs which have featured at the start of each of the book’s chapters point to one such: sona, or in singular...

21. Appendix A THE INCLUSION–EXCLUSION PRINCIPLE
(pp. 187-188)
22. Appendix B THE BINOMIAL INVERSION FORMULA
(pp. 189-192)
23. Appendix C SURFACE AREA AND ARC LENGTH
(pp. 193-194)
24. Index
(pp. 195-196)