Recent notes (1, 2) suggest the problem. Find necessary and sufficient conditions on a set of n(>3) real constants a1 …, anfor the inequality

The following two theorems are extensions of the Cayley-Hamilton theorem. The first gives the explicit expression for AN as a polynomial in A, where A is an arbitrary square matrix. The second expands the first result to the case of a matrix series.

In the year 1656 John Wallis published his Arithmetica Infinitorum, [1], in which he displayed many ideas that were to lead to the integral calculus of Newton. In this work we find the celebrated infinite product of Wallis which gives π,

Earlier in 1593, Vieta [2] found another infinite product which gives π

But, since Wallis does not mention it, we suppose that he was unaware of it. (Remarkably, these two seemingly different products are special cases of a more general formula [3].)

The December 1978 issue of the Mathematical Gazette [1] contains an elegant and humorous contribution from Anthony Hughes. His Note gives a recipe for determining the optimal spot from which to make a conversion attempt in rugby. Others then elaborated on Hughes' idea; see [2] and [3] in particular. In 1996 Isaksen [4] rediscovered Hughes' results while investigating the kicking of extra points in American gridiron. There are also a number of popularisations and summaries, in which the above results are presented, and in instances rediscovered. The ones of which we are aware are listed in the references; see [5, 6, 7, 8, 9 and 10].

In this paper I look at different aspects of mathematical reasoning, and argue that we need to make sure students of all ages engage in a range of mathematical reasoning, particularly given the evidence that teaching for reasoning is a powerful complement to teaching that is more focused on skills and procedures.

This article describes a new and general line-and-conic property, leading to a consideration of a ‘six-point’ circle and the related geometry of an interesting visual illusion.

In a well-known theory of infinite series, if a series converges it is taken to represent the number to which it converges, while if a series does not converge it is not thought to represent any number. Hidden behind this well-known theory, there are some intriguing examples of the representation of numbers by series which puzzled mathematicians of the 18 c. and 19 c. When taken individually the examples might be dismissed as being distinctly odd, yet because of the substantial degree of consistency which exists between them they have been studied extensively. Early disputes about the examples centred on the question of whether there is any sense in which the series could be regarded as representing numbers. Yet those who accepted such a series and used it in their work tended to ask the simpler question, what number does this series represent?

This note relates to the proof of the formula for the vector triple product (or continued vector product)

a ∧ (b ∧ c) = (a c) b − (a b)c, .....(1)

where the sign ∧ denotes vector multiplication and a dot denotes scalar multiplication.

If you visit the Manchester Museum of Science and Technology take a look at the sculpture in front of the museum. There are two large stainless steel objects, a sphere and an inverted cone. On the ground around the cone are variously shaped patterns of black and brown. However, when the patterns are viewed as reflected from the cone they take on the portraits of four famous scientists connected with the city of Manchester (see figure 1 on the previous page). This is an example of anamorphosis, or anamorphic art. Anamorphosis refers to an image or drawing which is distorted so that it can only be recognized when viewed in a particular way, or using a special device (like the cone in the above example). The word is derived from the Greek anamorphoun - to transform.