This is the story of a problem that began in an innocent way and in the course of time spread out in unexpected directions involving some interesting side-issues. It all began with the following problem which I constructed for the Mathematical Challenge competition run by the Scottish Mathematical Council.

John Gilder’s article on integer-sided triangles in the December 1982 Gazette [1] immediately suggests the question of what other triangles with integer sides have an angle which is a whole number of degrees. This is very simple to answer, for if the sides are integers, the cosine formula implies that the cosines of all three angles must be rational. Only angles of 60°, 90° and 120° fulfil this condition for the appropriate range of values, and it follows that, apart from the familiar right-angled triangle and Gilder’s case of the 60° angled triangle, the only other possible case is a triangle with an angle of 120°.

The criticism by T. B. M. Neil and M. Singer in The Gazette for October 1965 of C. P. Willans’ formula for the nth prime number is in my view quite unjust. Willans’ formula expresses two properties of the nth prime, namely that the number of numbers x such that 1 ≤ π(x) < n is two less than the nth prime, and that the nth prime is less than 2n. Willans gives a trigonometrical representation of π(x), the number of primes less than or equal to x. The analysis which Neil and Singer give is an exposure of the formula only in the sense that a clear proof is an exposure.

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.

Readers may be interested in an elementary exposition of the convergence properties of the sequence of iterates obtained by applying the Newton-Raphson process to an arbitrary polynomial. The coefficients of the polynomial may be real or complex. By restricting attention to polynomials we will give some precise results using the least possible analysis, while proofs for more general functions would necessarily require more high-powered methods. The arguments we will use require only an elementary knowledge of limiting processes.

The object of this note is to draw readers’ attention to a very new theorem in the Euclidean geometry of the triangle and to provide a straightforward Cartesian proof. This remarkable theorem, due to Lester, asserts that in any scalene triangle the two Fermat points, the nine-point centre and the circumcentre are concyclic. Lester’s original computer-assisted discovery and proof make use of her theory of ‘complex triangle coordinates’ and ‘complex triangle functions’ as expounded in, and . A proof has also been given by Trott using the advanced concept of Grobner bases in the reduction of systems of polynomial equations to ‘diagonal’ form. Trott’s work uses the computer algebra system Mathematica as an essential tool and he also provides an animation of the Lester circle as one vertex of the triangle is varied. Further information on the configuration is given in.

If you were to ask a random sample of students from your school if they liked mathematics, what do you suppose their responses would be? Almost certainly most of them would respond negatively and some would do so with considerable vigour. But what of the others? Some students do like mathematics and we might well ask why this is so? Do you suppose it is because these students realise that knowledge of mathematics is a necessary survival skill in today’s increasingly technological society? Surely the answer is no. Do you suppose it is because they realise that mathematics has many useful real life applications or that lack of knowledge of mathematics will effectively prevent them for entering college in most disciplines? Again the answer is almost surely no. Certainly mathematics is relevant and we should teach our students that this is so and we should teach them useful and appropriate applications as well. At the same time, it is my experience that student attitudes towards mathematics are much like they are towards exercise—in general, they do not exercise because it contributes to good health, but because exercising is usually done in the form of a game or a competition that they find attractive in and of itself. Similarly, what turns students on to the study of mathematics and provides the needed motivation for learning are the remarkable patterns and surprising results with which our subject is so replete. Successful teachers realise this fact and utilise it to maintain a high level of interest and enthusiasm which enables them at the same time to teach ideas which, while useful and necessary, may be more prosaic.