Julian Havil - Nonplussed!: Mathematical Proof of Implausible Ideas

Alice laughed: Theres no use trying, she said; one cant believe impossible things.

I daresay you havent had much practice, said the Queen. When I was younger, I always did it for half an hour a day. Why, sometimes Ive believed as many as six impossible things before breakfast.

Where shall I begin, she asked.

Begin at the beginning, the king said, and stop when you get to an end.

Lewis Carroll

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 authors case it is probable that in the long past the coin rolling around a coin puzzle provided Carrolls 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 the fixed coin, as shown in .

Is the rotated coin now head up or head down?

Within a random group of people both answers are likely to be proffered as being obviously true, yet one of them is false and a quiet experiment with two coins quickly reveals which. We must prove the fact though, and too much knowledge is dangerous here: fix on a point on the circumference of the moving circle and we have an epicycloid to consider (or, more precisely, a cardioid) and there could be hard mathematics to deal with. Alternatively, concentrate on the path of the centre of the moving coin and let us suppose that the common radii of the coins are r. During the motion, the path traced out by this centre is a semicircle, whose centre is itself the centre of the fixed coin and whose radius is 2r; the motion will cause the centre to move a distance (2r) = 2r.

Figure 1. A coin rolling around another fixed coin.

Figure 2. The situation simplified.

Now simplify matters and consider the moving coin rotating without slipping along a straight line of length 2r, the distance moved by its centre, as shown in . It is perfectly clear that it will have turned through 3600 and so be the right way up.

When it is first seen, the result is indeed surprising and the solution clever.

It is a suitable preliminary example as this book chronicles a miscellany of the surprising, with a nod towards the clever, at least in the judgement of its author. The choice of what to include or, more painfully, what to exclude has been justly difficult to make and a balance has been found which recognizes the diversity of the surprising as well as the large role played by probability and statistics in bringing about surprise: it is they and the infinite which abound in the counterintuitive; other areas of mathematics dally with it. To reflect all of this, the fourteen chapters which constitute the book are divided evenly and alternate between results which intrinsically depend on probability and statistics and those which arise in other, widely diverse, areas; one such is the infinite. To reflect these tensions further, this is the first of two such books, the second providing the opportunity to embrace what the reader may have considered as unfortunate omissions. Wherever it has been possible, the provenance of the result in question has been discussed, with a considerable emphasis placed on historical context; no mathematics grows like Topsy, someone at some time has developed it.