Mathematics at all levels depends on infinite processes. This is a “fact of life”. But, like those other “facts of life”, this one has traditionally been hushed up—especially when teaching elementary mathematics to beginners. Of course, in our liberated age, adult mathematicians are no longer embarrassed by infinite processes, and even discuss them openly. But it is not easy to see the connection between the way they discuss infinite processes (as in, say, an undergraduate analysis course) and those parts of elementary mathematics which depend on infinite processes, and which should provide the soil in which the beginner’s seedling intuitions about infinite processes can take root. So even today most children, adolescents, and their teachers must either make do with the traditional smutty versions, or work things out for themselves.

This article outlines a simple locus problem which leads to some interesting conclusions. It enables three elementary loci to be seen as special cases of a more general locus.

The problem might present a sixth-former with a challenging, open-ended investigation enabling him or her to practise some newly-acquired techniques in seeking the answer to an apparently simple question.

In 1972 Kenneth Arrow was awarded the Nobel prize for economics. Much of his work was in the theory and philosophy of decision making, in particular the theory of voting, and in this field “Arrow’s theorem” is a truly remarkable result. It is a genuine application to the social sciences of some elementary mathematics and as such I feel that it is a pity that it is not in the repertoire of more teachers. Hence this article.

Since the time of Gutenberg the volume of printed information has been doubling roughly every fifteen years. Only relatively recently, however, has the growth of knowledge reached a point at which it challenges human intellectual capacity. Our society has responded by a progressive specialisation of intellectual roles. Very broadly, the nineteenth century saw the development of specialism in distinct fields; the twentieth, the growth of specialism within those fields. To take just mathematics, it has been estimated that, after allowing for redundancy, our present knowledge fills the equivalent of something of the order of 60 thousand average-sized volumes. While at the start of this century, according to Ostrowski, it was expected that a mathematics graduate would be prepared to deal with any question in any branch of mathematics, by mid century Von Neumann was estimating that a skilled mathematician might be familiar with some 10% of the whole field.

The relationship in this article was discovered by Alan Tomkins and the proof supplied by David Pitt. The original inspiration was the statistical study of gambling systems—one method of attempting to win being to increase the amount staked each time you lose. The idea of this is that when you eventually win the amount won is sufficient to more than offset the losses on the preceding string of losers and put you back into the black. The snag is that this string of losers can leave you with insufficient funds to keep increasing the stake as necessary. The question which comes to mind is: in a given number of races, on how many occasions are we to expect a run of losers of a certain length?