I am looking forward with great enthusiasm to the arrival at advanced level of the GCSE student who has experienced a problem solving approach to mathematics through investigations. I hope examination boards are equally enthusiastic. Here for them is the opportunity to broaden their approach to syllabus content and methods of assessment. Can we hope that they are already working with this in mind? Advanced level mathematics courses, particularly in the options of applied mathematics, will need to nurture the good work of the GCSE courses and allow sixth-forms to produce a student capable of modelling real problems, in terms of sound and necessary mathematical techniques.

We all have experience of waiting in a queue. We have waited in the dentist's surgery for treatment, we have queued at the supermarket check-out with our trolley or stood in line at the Post Office to buy stamps. No doubt we have had frustrating experiences in such situations, but before we rush to criticise the discipline of a queueing system we should think about the mad stampede for the next bus that can occur in some parts of the world.

In 1742 Goldbach sent a letter to Euler which contained the conjecture that:

Every even number greater than 2 is the sum of two prime numbers.

It is easy to verify Goldbach's conjecture numerically up to some quite high limits but no proof has yet been found. I have long had the uneasy feeling that the conjecture is not about the prime numbers at all but would equally apply to any ‘sufficiently dense’ sequence of odd numbers. However a recent attempt to generalise the sieve of Eratosthenes has led me to regard Goldbach's conjecture with a good deal more respect.

During and after World War II, operational research teams needed new, computer-adaptable methods for optimization problems in industry, engineering, economics and the armed forces. The calculus methods developed in the sixteenth century to solve problems in astronomy, physics and geometry, even when they were applicable, were not always computer-adaptable. Among these new operational research methods is geometric programming, an optimization technique which in its beginning utilized the theorems in this article. The word “geometric” comes from its use of the geometric mean as well as geometrical concepts such as orthogonality and normalization.

The number of the Mathematical Association’s new headquarters in London Road, Leicester is 259.

This number has some quite interesting features. It is for instance the sum of the three squares

32 + 52 + 32 × 52,

and the sum of four cubes

23 + 23 + 33 + 23 × 33 and 13 + 23 + 53 + 53.