In Matters for Debate, David Singmaster (DS) discusses some topics relating to the use and misuse of decimal points: the use of a full stop in writing times of the day, confusion between the decimal point and a dot used for multiplication, and writing amounts of money.

In Catalan numbers with applications [1], Thomas Koshy gives a clear and accessible account of some of the important characteristics of Catalan numbers. The author also discusses the significance of these numbers for mathematicians, scientists, engineers and amateur enthusiasts alike. The general contents of his book are reviewed in [2]. Catalan numbers, which are named after the mathematician Eugene Catalan (1814-1894), have very interesting and sometimes surprising properties.

Catalan numbers occur in combinatorial mathematics, in particular, where they show up as a natural number sequence that arises in counting problems that usually involve recursively-defined objects.

Eperson's conjecture is that if n ⩾ 3 is odd then 3n2 can be written in at least two ways as a sum of three squares. We shall give a fairly elementary proof of this, from scratch.

Let q be a real or complex number with

Define for n ⩾ 1.

Space does not allow much room for “reporting” in The Mathematical Gazette, though that deficiency is greatly offset by the Newsletter. As it happens, however, two very contrasting items arrived while this Number was in the proof stage and so they are included, as typical of much else, to give some information about mathematical activities in general.

In teaching Analytical Geometry very little attention is often given to the interpretation of the form of an equation, although this may often convey a good deal of information; by grouping the terms of an equation in different ways, a variety of useful and interesting results may be obtained with very little labour.

Some suggestions as to the possibilities of this method may be afforded by the cases given below. A considerable number of results are obtained, purely by simple manipulation of the general equation of the conic section in point and line co-ordinates. Except in one instance, no reference has been made to infinite elements; if we include the notions of infinite points and lines, and of the circular points, the power of the method increases very greatly. The only knowledge that has been assumed is that which is quite fundamental in point and line co-ordinates, and the meanings of certain simple typical forms.

When considering ‘convergence’ many people think of number sequences or contexts arising from calculus. But there are also interesting phenomena of convergence – easy to visualize – arising in elementary geometry. Some of them are so elementary that they can be dealt with at school, for instance an example of iteration that is described in [1, p. 171f] (see also [2, p. 59], [3, p. 222f] and [4, p. 42ff]).

Which is your favourite formula in triangle geometry? Mine is definitely the formula for the area of triangle OIH . It is well known that the perpendicular bisectors to the sides of any triangle are concurrent at a point O (centre of the circumcircle), that the angle bisectors to the vertex angles are concurrent at a point I (centre of the incircle), and that the altitudes are concurrent at a point H. If the triangle is not isosceles, then these three points are all different and uniquely determine a new triangle OIH (see Figure 1), whose area can be expressed in terms of the sides a, b, c of the original triangle. I derived such a formula 20 years ago, and later found out that it had been studied a century earlier.