The first reference to the Problem Bureau in The Mathematical Gazette was in July 1928:

Few could cavil at the general views expressed about the place of mathematics in schools by the writers of the Newsom Report, as most of the ideas have been advocated in many Training Colleges for at least ten years. It is strange, however, to see that the section devoted to mathematics is the second part of Chapter 18 on “Science and Mathematics.” Does not this lend colour to the heresy promulgated by Hogben in “Mathematics for the Million” that the subject is to be valued principally, if not entirely, as the “Handmaid of Science”? Everyone will agree that “Mathematics has much to offer to boys and girls of average ability at the Secondary stage. It has a self-evident usefulness which can be a great advantage in arousing the interest of pupils.” Is it true that the only appeal to such pupils is through experience of the practical utility of mathematics in the world around them—the modern world of technology and science and commerce? Doubtless we need more and better mathematicians in all such activities, but they will not be drawn in any large numbers from these pupils. Numbers, shapes and patterns have an intrinsic fascination of their own that is independent of their practical value. For instance, it might be argued that the fact that “all numbers divisible exactly by 9 have a digit sum that is a multiple of nine” may be of some practical value—it was the basis of the process of “casting out the nines” that was formerly advocated as a quick check on the result of a calculation. Yet the pattern of the “nine-times table” is of aesthetic value as well as an aid to memory. Is the test to discover whether a number is a multiple of 11 of much practical value? Lewis Carroll used it to devise a method of dividing by 11 that depends solely on addition and subtraction, but its real value is the insight into number relations that is gained from demonstrating its validity: it certainly has an appeal to most people as a kind of “mathematical magic.” Do you know the test for discovering whether an amount of money expressed in shillings and pence is divisible exactly by 11?—just add the number of shillings to the number of pence, and if their sum is a multiple of 11, the amount of money is divisible by 11. There is an equally simple test for an amount of money to be exactly divisible by 13—can you find it? Experience shows that facts such as these stimulate the interest of those who dislike arithmetical calculations, whether young people or adults: the film-strip “Alice in Numberland” was designed to undermine the prejudice that many women students have against mathematics, by showing Lewis Carroll’s delight in puzzles and paradoxes, and his humorous approach to Symbolic Logic. Even in Secondary Modern Schools at all levels it should not be forgotten that mathematics can have an “entertainment value” comparable to that of music or any other form of art, if presented in the right way.

One approach to investigating the structure of a group is to examine the family of subgroups of the group. In particular, one often looks for normal subgroups and considers things such as direct products and composition series. However, fruitful investigation usually involves establishing that certain subgroups do, in fact, exist. Hence it is important to find conditions which will guarantee the existence of subgroups. This note indicates the usefulness of one such set of conditions—the Sylow theorems for finite groups.

In The Gazette for February 1966 (Vol. L, No. 371) Mr. C. A. Parrack introduced cliques to serve as counter-examples for groups. The purpose of this note is to show how cliques are related to groups. This leads to the classification of cliques in terms of commutative groups, and to considerable simplifications in clique calculations.

Among the greatest advances in knowledge this century one would include relativity, quantum mechanics and the explanation of hereditary replication in living things by the structure of the DNA molecule. But it would not be absurd to include in this category a discovery which is not at all well-known and goes back to 1930, namely, that logic and mathematics as we know them have undecidable propositions.

The first section of this paper deals with the Note 3104, A problem in differential geometry, by J. K. Brunton in The Gazette for October 1964. The second section was prompted by it.

The criticism by T. B. M. Neil and M. Singer in The Gazette for October 1965 of C. P. Willans’ formula for the nth prime number is in my view quite unjust. Willans’ formula expresses two properties of the nth prime, namely that the number of numbers x such that 1 ≤ π(x) < n is two less than the nth prime, and that the nth prime is less than 2n. Willans gives a trigonometrical representation of π(x), the number of primes less than or equal to x. The analysis which Neil and Singer give is an exposure of the formula only in the sense that a clear proof is an exposure.

Since all truth functions in two-valued logic may be represented by polynomials in the arithmetic of residues modulo 2, the question which R. Sibson raises in note 3127 may be expressed in the following form : Find all polynomials Φ(p, q) such that

identically, where p & q is a commutative polynomial which vanishes only when p = q = 0, and p & p = p. Taking p = q = r in (1) we find