Suppose that we open a telephone directory and select any column which consists entirely of personal names. The last digit of each telephone number would seem to have no preferred value. For example, the middle column on p. 417 of the 1970 Newcastle upon Tyne directory contains 109 names, and the frequencies for 0,1,…, 9 are as follows.

Every triangle can be used to tile or tessellate a plane because the 3 angles add up to 180° and so can be arranged to form a straight line. In a similar way every quadrilateral will tessellate because the 4 angles can be grouped round a point.

The article by Margaret E. Baron on the historical development of logic diagrams (The Mathematical Gazette, LIII, May 1969) aroused my interest in the infeasibility of drawing a diagram extendable to n classes. The difficulty of following Venn’s suggestion of a comb-like structure is shown in the complexity of drawing such a diagram for five classes.

I read the article by G. Merlane on the Use of Matrix Methods When Solving Simultaneous Linear Equations, (The Mathematical Gazette, LIV, 390, December 1970, pp. 337-41), with considerable interest, but as an outsider (with only vague recollections of traditional mathematics in his own schooldays) who has taken a rather casual interest in the developments of school mathematics syllabuses in the last ten years or so.

After fifteen years of using and teaching applied mathematics, I was called on to teach a course in elementary differential equations. To refresh my memory I consulted several recent textbooks and observed three major changes in their content over the intervening period, namely a clear description of the domain in which solutions are valid, extensive use of single valued and bounded functions and the avoidance at all costs of infinity.

This study was suggested by C. S. Ogilvy’s “Tomorrow’s Math”, which poses: “A weighing problem which has been going the rounds of some mathematical communities is the problem of the balls. Given n distinguishable balls, of which it is known that no two have the same weight, it is required that they be arranged in order of magnitude by weighings of one against another in a pan balance scale (no weights). What is the smallest number of weighings, as a function of n, which will always suffice? What strategy (procedure) does one adopt to achieve this minimum? Some assortments (depending on luck) will be more quickly ordered than others. Will the best strategy automatically minimize the expected number of weighings if the original ordering is random?”

In a previous issue of the Gazette J. G. Colman gives a formula for the nth derivative of the composite function f[g(x)], states that this is proved by induction and asks for a more direct and illuminating proof.

It is possible to project the four-dimensional hypercube γ4 into three dimensions in a symmetrical manner so that the edges lie mainly along the edges of a rhombi-dodecahedron but with two of the vertices coincident at the centre of the figure.

As A. M. Cohen’s paper has reminded me, Euler’s transformation for summation of the slowly convergent alternating series

is commonly derived by the following method, without enquiring into its validity which is distinctly dubious.

For all that geography, and economics, and sociology,… are becoming more mathematical, the fact remains that in the secondary school the mathematics teacher has far more points of contact with his science colleagues than with any other. Historically and in current practice mathematics and science march together—mathematics offering the language and the generality, science furnishing the motivation and feeding the imagination. Both are concerned with the identification of patterns, and progress by putting them to use.