Illustrative examples which draw on everyday experience are particularly useful in the classroom. For this reason, sport can be a good source of examples. We present four illustrations of the ways that mathematics arises in different sports. The ideas should be understandable to those studying A-level mathematics.

To all who live south of Watford, London is doubtless the hub of the solar system, as Boston State House was to Oliver Wendell Holmes. Asked for greater precision, they might opt for 10 Downing Street, Charing Cross, or Threadneedle Street according to taste. Most people I have asked “where is the centre of Britain?” plump for Meriden, but that answer would not commend itself to the inhabitants of Scourie. And if you should chance to be in Allendale Town on New Year’s Eve (assuming you can find the way to it), you will be in no doubt that this is where it’s at, and you have reached the roistering heart of Britain. But to give a more reasoned and accurate answer is an investigation well within the scope of a school class equipped with an O.S. map and a micro- or mini-computer.

Last Christmas holidays I stood on a point of great significance in the geography of Britain. The summit of Sgurr nan Ceathreamhnan (that’s approximately Skurr nan Kero-anan) at the head of Glen Affric with grid reference NH057228 is precisely on the watershed of Britain. Water falling to the west of the summit goes over the falls of Glomach and runs to the Atlantic at Dornie. To the east all the rivers drain into the River Beauly and into the North Sea at the Moray Firth. The watershed forms the source of a number of interesting problems at various levels of difficulty.

A number of authors have considered the following (and related) problems:

In the March 1981 edition of Mathematics Teacher Lee Markowitz gives the following full set of solutions (though not the considerable calculations involved):

(5, 12, 13) (6, 8, 10) (6, 25, 29) (7, 15, 20) (9, 10, 17).

However, in many formulae involving the triangle, it is the semi-perimeter rather than the perimeter that arises naturally.

Here are two fascinating topics in mathematics. The expansion of a number as a continued fraction, a heritage of the “old-fashioned” algebra curriculum, is happily enjoying a revival at present. The idea of a function having fixed points is so diffuse as to find applications in the most unexpected parts of mathematics and we shall see how it even has implications for continued fractions.

In his recent note “On buying postage stamps” in the June 1987 Gazette, John Gilder discusses the solution in integers of the equation

where x is the number of second class stamps (at the old rate of 12p), y the number of first class stamps (at 17p), and z the total cost in pounds. A simple algorithm, which can also be applied to equations involving three or more independent variables, is obtained from the following result.

Let k be a positive integer and a > 1. It is well known that

.

The shortest proof seems to be the following. By estimating the exponential function with only one term from its series we see at once that

.