The classical river crossing problem of the jealous husbands involves three couples who have to cross a river using a boat that holds just two people. The jealousy of the husbands requires that no wife can be in the presence of another man without her husband being present. This can be accomplished in 11 crossings (i.e. one-way trips). Tartaglia gave a sketchy solution for four couples but Bachet pointed out that this was erroneous and that four couples could not get across the river. In 1879, De Fontenay pointed out that four or more couples could cross the river if there was an island in the river and gave a solution for n couples in 8n – 8 crossings. Dudeney improved the solution for n = 4 and Ball noted that this gives 6n – 7 crossings for n couples.

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.

“Find the modulus and the argument of each root of the equation:

(University of London A level, part question.)

We were investigating iterative methods of solving the equation, using a micro-package (in BASIC) which performs complex arithmetic and displays the results in an Argand diagram. The facility for “doodling” which this provides led to some interesting conjectures and results, of which this note is an account.

The harmonic series seems to be a topic of enduring mathematical interest. Readers will be aware of the many neat methods which have appeared in the Gazette for showing its divergence but let us here investigate one of its discrete properties.

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.

(This article is the result of some work done by a first-year undergraduate, Tat-Sang Fung, at the University of Hong Kong. It went way beyond what his lecturer, Man-Keung Siu, had asked for, and he was so impressed that he submitted it to the Gazette. I have edited it a little, but the ideas are entirely Tat-Sang’s.)

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.

Suppose that an even number 2n of teams take part in a football league, and that each team is to play each other team once. Suppose also that the games are to be scheduled for 2n 1 days, each team playing one game on each day. How can a league schedule be produced? There are many ways of doing this, but one particularly simple method will be described.

Since the introduction in 1978 of the idea of a “common-core” for A-level mathematics, there was been a continuing debate over the structure and content of the subject at this level. Much of this debate remains pertinent in 1989: see, for example, [10], [15] and [28]. However, the setting-up of the Higginson Committee to review A-levels in March 1987 had the effect of putting much of the discussion specific to mathematics on the back-boiler. It is well known that the committee’s report of May 1988 recommended a wider subject-spread, roughly speaking on the model of the International Baccalaureate, and that its immediate rejection by the government created considerable furore in educational circles.

The following theorem about circles (which can be traced back at least as far as Archimedes) does not seem to be widely known and may be used to “pep-up” a discussion of circle theorems.

THEOREM. Let two chords of a circle meet at P and make a fixed angle θ (radians, say) with each other. Then the sum of the (opposite) arc lengths, L + L′, is independent of the position of P inside the circle (and thus, taking P at the centre, this sum equals 2rθ, where r is the radius of the circle).

There are some surprising connections between properties of solid convex polyhedra and the “nets” which can be used to make their shapes. These connections may be used as a starting-point for some interesting classroom investigations to “spot the pattern” or “guess and test”.

For a convex polyhedron, such as the pyramid shown below on the left, Euler's relation says that

Vertices + Faces = Edges + 2

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.

In a recent session of the Royal Institution Master Classes in Cambridge, the enthusiastic participants explored a computer graphics program, designed and written by Derek and Alan Ball. The program first draws a circle, and, given an integer N, marks N equally spaced points on it, these being cyclically labelled 1,2,…,N. The operator then inputs some rule, say n ↦ 5n + 8, and the program draws, for each n, the chord from the point marked n to the point marked 5n + 8 (modulo N). Before long, the children took N to be large and flower-like envelopes of the chords started to appear (as illustrated below). Here, we show how the equations of these envelopes can be obtained in a simple way with some general parameter k, and that for positive k they are all epicycloids, that is, curves obtained by rolling a circle around a fixed circle, the simplest one being a cardioid. A similar analysis shows that for negative k they are hypocycloids (see for example E. H. Lockwood’s Book of curves, C.U.P. 1961) except when k = 1 when parallel chords occur.

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.