Eddington's ‘liar problem’ received its first truly public airing in his book New Pathways in Science [1], although it enjoyed a ‘prehistory’ (beginning with a mock examination paper) more fully detailed in his later discussion [7]. As posed, it reads:

So, for example,

x appearing 3 times,

and x appearing 5 times.

The hyperexponential function is then defined as

for those values of x for which this limit exists. Since an infinite number of operations are involved, we clearly need to study carefully the question of convergence.

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!

For any three points X, Y, Z, let ⊙XYZ denote the circle through X, Y, Z (the circumcircle of ∆XYZ) or, if X, Y, Z happen to be collinear, the line XYZ. (We shall often regard lines as special circles, circles of infinite radius.) This paper is about the following theorem, and extensions of it:

Theorem 1: Given ∆ABC and a point P, reflect ⊙PBC, ⊙APC, ⊙ABP in the lines BC, AC, AB respectively. Then the three reflected circles have a common point, Q (see Figure 1).

I do not know if this theorem is new, but I have not come across it in the literature. The reader is invited to prove it by angle-chasing, using circle theorems: let two of the reflected circles meet at Q and then prove that this point lies on the third reflected circle. This method is rather diagram-dependent, and does not seem to lead to the extensions of Theorem 1 referred to above in any very obvious manner, so we shall adopt a different approach.

During an investigation on some Diophantine systems, we were led to consider the following set

where x, y, a, b, c are indeterminate integers.

From elementary algebra we know that, if they exist, x and y are the roots of the quadratic equation

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.

The fact that an abelian group can be associated with an irreducible cubic curve has been known for a long time, and many results of great beauty are known about such groups. Ever since Wiles' proof of Fermat's Last Theorem, some of these results have become well known even outside the mathematical world.

It may come as a surprise to the reader that there is a natural way to associate an abelian group with a much better known and simpler class of curves – the conic sections! In this article we explore some properties of these groups.

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 classical dynamics, Newton’s laws hold in all inertial frames. In particular, the work done by a force along the path traversed by a particle equals the change in kinetic energy of the particle. However, velocities and paths are frame dependent. Thus we investigate the work of a force in two inertial frames. We see that work is frame dependent. Furthermore, we determine the relation between work in two inertial frames.

An order-n Latin square contains numbers, each of which is one of a set of n real numbers, , arranged in the form of an n × n matrix, in such a way that each row and each column of the matrix contains all n numbers. Euler (1707-1783) was the first to study the properties of Latin squares and they have been the focus of continued attention since. Studies of Latin squares naturally lead one to elements of group theory and of matrix theory. As will be shown in this note, both of these features may offer interesting investigative opportunities for classroom discussions of the permutation group on three symbols and of the algebra of the associated permutation matrices.

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 the second named author presented some results and problems concerning interlinked loops. These were ‘inspired’ by the well-known Borromean rings – three loops with the property that they are interlinked, but if we cut and remove any one of the loops, then the remaining two loops are not linked and so can be separated.

To conform more closely with standard terminology in this area we shall here use the term ‘link of n components’ to denote such a set of loops (rather than linkage). In all the cases we consider, the individual loops that make up each link are unknotted.

A geoboard is a flat piece of wood with nails driven in at integer lattice points, and polygons are constructed on the geoboard by placing rubber bands over the nails. At a workshop for teachers, held at Stellenbosch in South Africa, and organised by the African Institute for Mathematical Sciences Schools Enrichment Centre (see http://aims.ac.za/aimssec), it was conjectured that if two polygons on a geoboard overlap, then the the area of the overlap is rational. It is not difficult to prove this but, as we shall see, the ideas in the proof go far beyond the original problem and may be of interest to some Gazette readers.

This paper is motivated by the work of V. Dubrovsky [1] who described an n-dimensional cube. It is our aim to build a fractal, based on such a cube as well as to derive a formula for a general volume of that fractal. Two special cases n = 2 and n = 3 are investigated in detail.

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.

The concept of isogonal conjugate points and isotomic conjugate points relative to a given triangle appears to have been introduced by Casey in 1888. Since then these topics have been treated in many important books, such as Johnson Coxeter and Greitzer and more recently Honsberger. The idea originated from the fact that the isogonal conjugates of the points on a fixed line form a circumconic of the triangle involved. That certain pairs of points are isogonal conjugates is of considerable theoretical significance. For example, that the symmedian point K in a triangle is the isogonal conjugate of its centroid G is an important result in understanding the significance of the symmedian point. The concepts involved, though well known to enthusiasts of triangle geometry, deserve additional publicity.

Recently, when solving a jigsaw puzzle, I noticed that many of the pieces were symmetric in some sense; they had either reflective or rotational symmetry. This made me wonder if it was possible for all the pieces in a puzzle to be different shapes, and yet have some prescribed symmetry.

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