This edition of the Gazette is the first that bears my name as editor (although it will be several editions before the change-over is complete). So this is an appropriate time to pause and take stock. First, I would like to thank Douglas Quadling for his brilliant years as editor. No-one who has met him can fail to marvel at the enthusiasm and devotion which he has applied to his task; and no-one who reads the Gazette can fail to appreciate how the journal, under his guidance, has kept its sense of balance through almost a decade which has seen some considerable upheaval in the teaching of mathematics. Second, as a newcomer I really do need to know who reads the Gazette. You do, presumably; so you are the obvious person to tell me what you like or dislike about it, and to tell me about any particular aspects which you would like to be introduced/abolished/expanded/reduced. I will welcome your comments and do my best to steer the Gazette on a course which’

When a function in arithmetic modulo n is represented through a cartesian graph, the resulting set of points reveals little information about the form of the function (for example, see Fig. 1). It is not easy to discern connections between the points; a restriction to the values 0, 1, 2,…, n — 1 would have to be placed on x and y; and the fact that n — 1 is as ‘near’ to 0 as 1 is, is not apparent in the graph.

Formulae for Σl=1n ik when k is a positive integer are well known, and many methods for finding them have been described. This article gives an account of an approximate, but remarkably accurate, formula which I have found for the case k =1/2.

Nowadays no mathematician is surprised when he hears someone speaking about an n-dimensional space: in fact, work involving vector spaces and functions of several variables makes every student familiar with this concept.

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.

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.

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).