My theme is a Manifesto for Mathematics. What is the nature of a manifesto? It is a statement of intent; a statement of principles. It looks at the current position and sees what needs to be done. The best manifestos do not make detailed promises about what will be done and when in an exact form … for indeed circumstances change. However the principles should be enunciated clearly enough to be aware of what directions will be set. Manifestos are also compromises. In Wales, I am trying to get groups together to agree a way forward to put to the Welsh Assembly Government for mathematics. I am sure that my more idiosyncratic views will be well hidden. However this morning no such constraints face me, and my manifesto will be very much a personal statement of what I see to be the current position and the correct way forward. Please do not see this as an official view!

The Frog Problem is a well-known exercise in problem solving that involves a lot of reasonably deep mathematics that is perhaps of more interest than learning a new integration technique. I want to try to expose this mathematics here, not that it is within the scope of all school students but so that it is recognised and possibly offered to at least the brightest pupils.

A perfect number is a number which is the sum of all its divisors except itself, the smallest such number being 6. By results due to Euclid and Euler, all the even perfect numbers are of the form 2P-1(2p - 1) where p and 2p - 1 are primes; the latter one is called a Mersenne prime. Whether there are infinitely many Mersenne primes is a notoriously difficult problem, as is the problem of whether there is an odd perfect number.

In the classical Monty Hall problem you are a contestant on a game show confronted with three identical doors. One of them conceals a car while the other two conceal goats. You choose a door, but do not open it. The host, Monty Hall, now opens one of the other two doors, careful always to choose one he knows to conceal a goat. You are then given the options either of sticking with your original door, or switching to the other unopened door. What should you do to maximise your chances of winning the car?

In a previous note the present author has given an account of the conjugations known as isogonal conjugation and isotomic conjugation with respect to a given triangle and has shown how to generalise the idea of conjugation involving a pair of points so that P(l, m, n) is q-conjugate, by means of a construction involving a transversal q, to the point (p2/l, q2/m, r2/n) and he has also demonstrated how to perform the geometrical constructions involved. When there is a self-conjugate point at the incentre I one has isogonal conjugation, and when there is a self-conjugate point at the centroid G one has isotomic conjugation.

In an earlier paper in the Gazette the authors of define what it means for a matrix in a set M of n × n matrices to be prime, namely if it is not the product of two matrices in M, neither of which is the identity. They then showed that there are exactly two primes in the set M2 of 2 × 2 matrices with non-negative integral entries and unit determinant, namely

I recently stumbled on an identity (Theorem 2) that related two types of permutations. It looked like one of those serendipitous pieces of mathematics that led to a simple enumerative result; mostly when this happens it leads joyously to an enumerative proof that was ‘blindingly’ obvious. Except in this case, it wasn’t, I don’t think.

The concept of the ordinary binomial coefficient

can be employed to construct an interesting family of positive integers. Such a family was introduced around 1974 by W. Hansell using the triangular numbers , where we call them tribinomial coefficients since they are binomial coefficients for triangular numbers. To this end, first we define corresponding to and For example,

In the problem proposed by A. G. Ioachimescu in 1895, it is asked to be shown that the sequence , defined by , for each , is convergent and its limit lies between -2 and -1.

There have been given many generalisations and other results regarding Ioachimescu’s problem in the literature (see, for example, [2], [3], [4, Theorem 1, parts (a) and (b)], [5, problem P2, parts (i) and (ii)],).

The German version of Riemann’s Collected Works is confined to a single volume of 690 pages. Even so, this volume has had an abiding and profound impact on modern mathematics and physics, as we shall see. In fifteen years of activity, from 1851, when he gained his doctorate at the University of Göttingen, to his death in 1866, two months short of his fortieth birthday, Riemann contributed to almost all areas of mathematics. He perceived mathematics from the analytic point of view and used analysis to illuminate subjects as diverse as number theory and geometry. Although regarded principally as a mathematician Riemann had an abiding interest in physics and researched significantly in the methods of mathematical physics, particularly in the area of partial differential equations.