It is just twenty years ago that sixty school teachers (mostly heads of mathematics departments) and twenty representatives of industry spent ten days together at Trinity College, Oxford for an intensive exchange of information and ideas. They had as their theme the changes that had been taking place in the uses of mathematics, and their relevance for school teaching. The conference was a seminal one in the development of British mathematical education; it made important personal contacts, it widened horizons, and it could also be described as the great-grandfather of SMP (the grandfather and father being the conferences which followed in 1959 and 1961 at Liverpool and Southampton). With the emphasis nowadays on the importance of keeping school mathematics in touch with the needs of industry, it is germane to remind ourselves of the influence which industrialists have exercised—personally, financially and mathematically—on the development and dissemination of new curricula in this country. Members of our Association may not always appreciate the contribution of its Schools and Industry Committee in keeping the dialogue going.

A teacher can make a great deal out of some of the questions which pupils and friends put to him from time to time. In an article in Mathematics in School a few years ago [1] I listed a number of such questions and discussed one, which involved a geographical application of matrices, in detail. The present article continues this theme, but its main aim is to stimulate discussion of the place of “mathematical modelling” in the curriculum. For it is felt by some that, in spite of the opportunities offered by (say) linear programming, mechanics and probability, insufficient emphasis is placed on modelling in the secondary school. It has been said too that we have not been notably successful in finding good applications of topics new to our syllabuses and incorporating them into our teaching [2].

There seems little doubt that the proportion of the mathematics sixth form who include traditional applied mathematics as part of their studies is steadily decreasing, and that this decline in its turn affects both the popularity of, and the ability of students to cope with, applied mathematics in universities. It is evident that fewer sixth formers are taking the ‘double mathematics plus physics’ route to reading mathematics at degree level; the combinations taking its place are many and varied, but often include statistics, for example, instead of applied mathematics, and economics or geography or a language instead of physics. Even where some students take so-called double mathematics, the syllabus followed often gives statistics as an alternative to applied mathematics, and an increasing number of schools are teaching this option.

Mathematics has had a profound influence on economic theory within the last 25 years, and many well known economists started their higher education in mathematics. Alfred Marshall and John Maynard Keynes, for example, were both Wranglers at Cambridge; we also know that Adam Smith—whose Inquiry into the nature and causes of the wealth of nations was published just over 200 years ago—studied logic and mathematics at Glasgow; and, more recently, some scholars have come to regard Karl Marx as one of the ‘giants’ in the history of mathematical economics.

Matrix models are used extensively in demography and population biology. The example discussed here shows how a simple 2 × 2 matrix may prove of use in the study of bird populations. Teachers may find it an interesting illustration of the application of matrices. The knowledge of matrices that is required to appreciate the model and investigate it numerically may be summarised as (1) the expression of two linear equations as a matrix equation, (2) the multiplication of matrices. A deeper understanding of the behaviour of the model comes from the use of eigenvalues.

The idea of a probability generating function seems to be a difficult one for sixth formers, and this may well be because the formalism is introduced too abruptly. The t or z or whatever letter is used in a p.g.f. has to be explained as having “no physical significance”, to quote from a recent M.A. booklet on sixth form probability and statistics. The following gentler approach I have found to lead to better understanding on the part of ‘lower fliers’, and it has the advantage of drawing attention to an underlying structure.

The idea of this article is to find a ‘simple’ expression for the number of circuits of length 7 in the directed graph illustrated; that is, a path along

the arcs of the graph which passes through each vertex exactly once and returns to the vertex from which it started.

The Witch of Agnesi (or versiera) is a curve named for Marie Agnesi who first studied it. It may be defined geometrically as follows: In a circle with centre at and diameter a (hence tangent to the x-axis at the origin), a secant through the origin is drawn, intersecting the circle in A and the line y = a in B. The intersection of lines through A parallel to the x-axis and through B parallel to the y-axis determines a point P. The Witch of Agnesi is the locus of the point P as the angle the secant makes with the x-axis assumes all values from 0° to 180°.

The method described in this note is, like other standard approaches, based on the “Euclidean algorithm”, but a matrix format is used (assuming the coefficients have no common factor) to reduce the equation ax + by = c to a form such as

or a similar equation with matrix (k l),(k −l) or (−1 k). The following lemma is easily verified.

The following ‘paradox’ was produced by one of my fourth year (14-year-old) pupils in an attempt to solve a quadratic equation. (The fourth years have no knowledge of techniques for solving quadratics, nor of course of complex numbers.)

During a study of the probabilities involved in an experiment with dice I needed to determine the frequencies of appearance of certain categories of display when six dice are thrown simultaneously. The theory of this led to the polynomial which is the subject of this note.

A computing colleague asked me recently if I could devise an algorithm for obtaining the ith permutation in an alphabetical list.

Suppose that the n! permutations of n distinct letters A, B, C,..., N are listed in alphabetical order. The problem is: Given i (where i = 1,2,...,n!), find the ith permutation.

In the article Problems, solved and unsolved in the Gazette for March 1976, and in the postscript to its sequel in December 1976 (60, 1–11 and 284–291), proofs are given that

is integral when a and b are positive integers. The search for a combinatorial proof was unsuccessful.

In Note 60.3 (March 1976) Mr. Gubby revived the old problem of the two ladders. If (Fig. 1) the lengths of the ladders AD, OB are m and n, and they meet at a height p above the ground, then the problem is to find OD, the distance apart of the walls, in terms of m, n and p.

The purpose of this note is to show how two first integrals of the equation of motion of a particle moving under the action of an inverse square central force can be obtained very simply using vector techniques. In fact it is shown that, amongst all central forces, only the physically important inverse square force can be treated using the technique given here, thus demonstrating one facet of the mathematical simplicity of that force.