No one can possibly welcome more cordially than I do the widening of the school curriculum in mathematics that has taken place during the last twenty years. Even I can remember the days when ‘Conics’ and ‘Mechanics’ were the privilege of a select few, and only a prodigy was initiated into the mysteries of the Differential (much less the Integral) Calculus. All this is changed now, and it is fairly safe (at least that is my experience) to assume that a boy who has won a scholarship has learnt something about the Integral Calculus at school.

Recent years have seen a marked increase in the tempo of the hunt for digits of the decimal expansion of π as shown in the table below.

Groups have for several hundred years been used in the composition of long peals in change ringing, as was mentioned on p. 399 of the Gazette. A brief outline of their use for the non-ringing mathematician may be of some interest.

Numerical integration, including the numerical integration of differential equations, is becoming increasingly necessary in technical applications. For such purposes, when a very high order of accuracy is not required, formulae which use successive values of the function to be integrated are more convenient than those using differences. Individual authors have given formulae of this type from time to time but a systematic list is not known to the present writer, and would seem to be of service.

The present list gives formulae of four types : (I) formulae for evaluating definite integrals ; (II) formulae for forward integration ; (III) formulae for integrating over a subrange ; and (IV) formulae involving values of both the function and its integral. The leading term in the error series is given. Finally, some comments are made upon the formulae and their uses.

I should like to say straightaway that collecting cigarette-cards is not my hobby But recently the manufacturers of the brand of cigarette I smoke started to issue an attractive series of cards and I said to a friend, “I think I shall save these cards until I obtain the complete set of fifty.” He replied, “About how many packets do you think you will have to buy before you get the set?” And this raises an interesting problem in probability which I do not recollect having seen before.

As the name suggests, the Multinomial Theorem is an extension of the Binomial theorem, and it was when I first met the latter that I began to consider the trinomial and the possibility of a corresponding Pascal's triangle.

My mathematics master suggested that I construct the triangle myself. It became apparent that such a triangle may not be drawn, as the result is a pyramid or a tetrahedron where the apexes of the tetrahedron are occupied by the coefficients of xn, yn, zn in the expansion of (x + y + z)n.

This may be shown by forming expansion coefficient triangles for a trinomial of power 0, 1, 2, 3.

The intersection of two circular cylinders of equal radius is not only of mathematical interest but also has application in both engineering and architecture. The joining of pipes of circular cross-section at a variety of given angles is an obvious example. The Romans and Normans, in using the barrel vault to span their buildings, were familiar with the geometry of intersecting cylinders where two such vaults crossed one another to form a cross vault. Larger numbers of equal intersecting cylinders arise in the following way.

First it may be necessary to remind some readers of what is meant by the representation of a group by a Cayley graph. Each element of the group is represented by a point, the points being connected by lines (or edges) each of which represents a generating operation of the group. Suppose, for example, that x and y are two elements of a group, while another element;? is taken as a generator, and that these are connected: xp = y. (Throughout this article we shall take the identity element of a group to be 1, inverses, to be indicated x-1, while xp means “first x, thenp”.) This would be graphed:

In this paper a magic cube of order n is a 3-dimensional matrix Qn = {qn(i,j,k); 1 ≤ i,j,k ≤ n} containing natural numbers 1,2, ... n3 such that the sums of the numbers along each row (n-tuple of elements having the same coordinates on two places) and also along each of its four great diagonals are the same. This contrasts with a previous article [1] which considered magic cubes with no requirement for the diagonal sums.

1. Background

From the end of the 17th century until the 1980s, most approximations to the value of π made use of Gregory’s series:

which converges for values of tanθ between -1 and +1. Since , the simplest, and most slowly convergent, series of this kind is

Writing x = cotθ, the series becomes

Expressions for π/4 have been devised which involve the inverse cotangents of larger numbers, causing the series to converge much faster; of such expressions (referred to herein as ‘identities’) probably the most celebrated is ‘Machin’s Formula’:

Problems about the American city road-system with its streets running east and west and its avenues running north and south have appeared from time to time in examination papers and are interesting because they give relations between binomial coefficients which are harder to get by straightforward algebra. After illustrating some of the methods by which difficult-looking results can be obtained simply, I shall obtain a simple-looking result with difficulty and ask for an easier way of getting it.

Ever since I was a schoolboy, my favourite polynomial has been

It is a wonderful polynomial, with lots of beautiful and useful properties which, alas, do not seem to be known to the present generation of mathematical students. To remedy this, we collect together some of the known properties of this polynomial and present some additional properties which appear to be new.