This paper contains many opportunities to investigate, conjecture and verify results by writing simple computer programs, and the reader is strongly urged to participate in this way throughout.

For each positive integer n, let F(n) be the sum of the squares of the digits of n; for example,

One of the pleasant aspects of research into mathematical history is the way in which existing material, passed over in the conventional account, may allow us not only to establish the bare, if unexpected, fact of a priority but, more importantly, to assess the significance of major currents in mathematical thought with greater precision. Briggs’ partial discovery, in anticipation of Newton, of the general binomial expansion,

is an excellent case in point.

If from a point, regarded as a centre of inversion, a radius is drawn on which are two points at distances r1 and r2 from the centre, satisfying the condition r1r2=r2 = constant, the points defined by r1 and r2 are said to be inverses one of the other.

In tackling a problem, we needed to show that a certain 3 × 3 determinant is non-zero. This determinant could be expressed in terms of the symmetric, homogeneous function N = Σx.Σx2 + 6xyz - 2Σx3 of three variables, where, for example Σx2 = x2 + y2 + z2. We first looked into several classical books, to see if there was a relevant inequality. There was not, nor do they contain anything other than ad hoc techniques for dealing with the inequalities they give. In this paper we give a general method for dealing with a certain class of such inequalities; it might be instructive for undergraduates, partly to exercise them in applying a moderately complicated algorithm, and partly to show an application of 2- and 3- variable Calculus.

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.

It is a well-known fact that the number of hours of daylight varies with the time of year, except at points on the equator. It is sometimes thought that at latitudes between the Arctic and Antarctic circles the variation takes the form of a sine curve. However there is no theoretical basis for this idea, although it is true that a sine curve may give quite a good approximation.