Shortly before the start of my term as MA President, a colleague asked me what my theme was going to be for the year. “Do Presidents have themes?”, I asked. He didn’t know, but thought it might be a good idea. As it happens, a perfect theme was handed to me on a plate. Planning for the 2008 Annual Conference was already underway when I joined Council, and it had already been decided that it was to be called ‘Joined Up Mathematics’.

What is the area of the shaded region between the tyre tracks of a moving bicycle such as that depicted in Figure 1 ? If the tracks are specified, and equations for them are known, the area can be calculated using integral calculus. Surprisingly, the area can be obtained more easily without calculus, regardless of the bike’s path, using a dynamic visual approach called the method of sweeping tangents that does not require equations for the curves.

This is the second of two articles presenting new results which extend the Soddy circles of a triangle to Soddy spheres of a special class of tetrahedra. In Part 1 [1], after a brief résumé of the Soddy theorems in two dimensions, we stated the corresponding theorems in three dimensions, discovered partly as the result of computer investigations. These threedimensional Soddy theorems are valid only for a special type of tetrahedron - the four-ball tetrahedron - and we went on to prove some basic results about such tetrahedra.

Are there solutions of the equation x2 + 1 = 0 ? Carl Fredrich Gauss (1777–1855) conjectured that there was a solution and that it was the square root of - 1 . But since the squares of all real numbers, positive or negative, are positive, Gauss introduced a fanciful idea. His solution to this equation was , which he named i. He integrated i with the real numbers to form a set known as , the complex numbers, where each element in that set was of the form a + bi, where a, . Gauss illustrated this on a graph, the horizontal axis became the real axis and represented the real coefficient, while the vertical axis became the imaginary axis and represented the imaginary coefficient.

We first present a general property of nice polynomials, which leads to an important modification of the concept of equivalence classes of nice polynomials. We then establish several properties of nice symmetric and antisymmetric polynomials with an odd number of distinct roots. We give a complete description of all nice symmetric and antisymmetric polynomials with three distinct roots. We then apply our results to nice symmetric sextics and octics. As an important application, all the properties we have established have dramatically increased the speed of a computer search for examples, allowing us to discover the first examples of nice symmetric and antisymmetric polynomials with five distinct roots and the first two examples of nice polynomials with six distinct roots.

Pascal’s triangle is the most famous of all number arrays - full of patterns and surprises. One surprise is the fact that lurking amongst these binomial coefficients are the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers which arose in the Hindu studies of arrangements and selections, together with the Fibonacci numbers from medieval Italy. New identities continue to be discovered, so much so that their publication frequently excites no one but the discoverer.

Let Q be a convex quadrilateral ABCD. We denote by TA, TB, Tc, TD, the four triangles BCD, CDA, DAB, ABC, respectively

The barycentres (or centroids), orthocentres, incentres and circumcentres of such triangles determine other quadrilaterals in their turn that we call the quadrilateral of the barycentres, of the orthocentres, of the incentres and of the circumcentres, respectively. We denote these quadrilaterals by Qb, Q0, Qi, Qc, respectively.