The reader will be familiar with the golden rectangle, i.e. that rectangle which leaves another rectangle of the same shape when the largest possible square is cut from it (Fig. 1). In this article the effect of successively cutting squares from rectangles of other shapes will be investigated.

The increasing use of simple computing languages in schools gives an opportunity for rationalisation of our existing mathematical notation as follows:

It is time we abandoned “÷”, which is used in primary schools and causes great difficulties later on when children have to cope with the fact that or, worse still, If we used “|” from the start, what could be easier than (Nor is there a case for restricting ÷ to ‘sharing’ situations, since + and − are used for both natural numbers and rationals.)

A typical sixth form book which is introducing vectors may approach the subject in the following manner. Of necessity, this particular introduction is rather condensed, but the reader will almost certainly recognise certain key phrases and be familiar with the general spirit of what is intended.

Any simple transformation that leaves the origin unchanged corresponds to multiplication by a 2 × 2 matrix which can generally be written down at sight. Transformations such as translations, under which the origin moves, cannot be represented in this simple manner. However, by moving into three dimensions this deficiency can be made good. Furthermore, the exercise gives useful experience of thinking in three dimensions and a useful addition to the repertoire of matrices.

Boys who have learned to find π using jam jars and string are often mystified to learn that the number has been calculated to thousands of decimal places: the infinite series and products usually quoted cannot be justified without analysis.

This article is about a method whose derivation involves only elementary algebra and trigonometry (including square roots and Pythagoras), and whose use is a feasible proposition even when only four-figure tables are available for the square roots.

If on the sides of a triangle ABC equilateral triangles are constructed, inwardly and outwardly, and their circumcircles drawn, the 6 circles will in general have 11 points of intersection: A, B, C; 2 points on each of the 3 sides; a point F common to the circumcircles of the outer triangles, and a point N common to those of the inner triangles (Fig. 1).

This article uses complex numbers to investigate the figure, and has as its main object the establishment of an extremal property of N.

B. J. R. Bailey’s article in Gazette LVII, 169–174 (October 1973) worried me a little, because it seemed to suggest support for a widespread misapprehension, that mathematical statistics differs from other branches of applied mathematics in requiring a good deal of subjective argument. Every application of mathematics involves judgement, of course, but statistics does not really go beyond other applications in this respect.

Several modern mathematics courses contain a description of the 17 distinct ‘wallpaper patterns’. Others contain the definition of “group” and “isomorphism”, together sometimes with a vague statement that these concepts can be used to justify the fact that there are precisely 17 patterns. But neither the passive contemplation of wallpaper patterns, nor the passive contemplation of abstract definitions, is mathematics: the latter is above all an activity in which definitions are used to obtain concrete results. For this reason I have often been asked by teachers what is needed to give a rigorous proof that there are precisely 17 patterns.

One of the oldest and best known of chessboard problems is to place the largest possible number of similar pieces on the board such that no two of these pieces are attacking each other. In his book Amusements in mathematics [1], Dudeney considered this problem on a generalised square chessboard containing n 2 cells, and proved that for rooks, queens and bishops this maximum number is equal to n, n and 2n – 2 respectively.

In a course on Greek Mathematics for our Classical Sixth some years ago, I noticed a connection between Theon’s and Hero’s sequences of rational approximations to √2 (the “side-and-diameter” numbers). A footnote in Heath [1] refers to a proof published in German in 1886, but I recently came across a generalisation which may be of some interest.

The popular definition of a “straight line” as “the shortest distance between two points” is one of the few pieces of mathematical jargon in general circulation in the English language. This is unfortunate, for it is not a good definition, and one may well wonder who originated it. Euclid is not to blame, for he defined a straight line as “that which lies evenly between its points”. (This is indeed cryptic, and Sir Thomas Heath, in a lengthy analysis of the original Greek, had to confess that “the language is...hopelessly obscure”.)