In many classrooms now a mention of Pascal will produce vague memories of “that triangle of numbers” (Table 1). Some children may even be able to reproduce it. If they can provide an application it is likely to be about numbers of routes or probability. At a later stage the better mathematicians will associate it with the expansion of (1 + x)n, where the row number gives the value of n and the column number the power of x, and the binomial expansion generally. Let me present you with two more tables which I have dubbed Super-Pascal because the rules of compilation are reminiscent of the rule for the triangle above but a little more complicated.

Among the first examples of groups encountered in group theory are the symmetric and alternating groups and the general linear groups. The purpose of this article is to describe these groups and to describe two unusual functions between groups of this type, in a language which can be understood by those with only an elementary knowledge of group theory and no knowledge of linear algebra or projective geometry.

Provided one of the integers m and n is even, it is obvious that an m × n rectangle can be covered with 1/2mn dominoes, each domino covering two unit squares. If some of the dominoes are oriented in each of the two possible ways, it is possible that there may be no continuous straight line of domino edges, either horizontal or vertical, across the rectangle: in other words, each of the (m — 1) + (n — 1) fault lines on the underlying grid is crossed by at least one domino. Such rectangles are called fault-free, and are mentioned by S. Golomb in [1] where the concept is attributed to R. I. Jewett. An example with m = 6, n = 8 is shown in Fig. 1.

Prior to the advent of the calculator, divisibility tests were often studied and utilised. Some of these tests were simple and easily remembered; others, more cumbersome and rarely utilised in either practical or theoretical work. The purpose of this article is to present a simple and general test for divisibility which is of applicable and theoretical interest.

The Norwegian mathematician Abel wrote in 1826 [1] “On the whole, divergent series are a devilry, and it is a shame to base any demonstration upon them”. At the other extreme Heaviside, a British physicist and engineer whose unorthodox methods raised many mathematical eyebrows, is quoted [2] as having said “The series is divergent; therefore we may be able to do something with it”. Heaviside may be accused of exaggeration but Abel, though his attitude was very reasonable in the absence of a proper theory of divergent series, was somewhat over-zealous in his total condemnation of the use of such series. Divergent series can be useful and in some cases are of far greater practical value than their convergent counterparts.

The two articles which follow, which deal with similar themes in quite different ways, arrived on the editorial table within a short time of each other. Since they are complementary, it seems appropriate to present them here under a common heading.

Collecting the free picture-cards in packets of cigarettes or tea has always been popular. The equivalent up-to-date example is the gift of a picture—card of a football team with each purchase of petrol. Many a parent has been asked the question ‘how soon will we get the set’. Let us state a fairly general problem based upon this idea of collecting a set. Our problem states:

From any two given functions of x, f (x) and g(x), we can (if domain and range agree) construct their compositions f (g(x)) and g(f (x)). Thus from x2and tan x we construct (tan x)2 and tan (x2). I started this investigation by wondering about the inverse situation, where, given a function h(x), we are interested in finding all possible pairs of functions f(x), g(x) whose composition f (g(x)) yields h(x).

Given a square matrix A with real or complex entries, there may not exist an invertible matrix S such that S-lAS is diagonal. For instance,

cannot be diagonalised in this way (the contrary assumption leads by easy direct calculation, to a contradition); however, this matrix is already upper-triangular.