In this article I review some of the more advanced results concerning cyclic quadrilaterals, including a number that are new to me and possibly, therefore, to readers. Proofs of theorems are not given, but I give a clear indication of the methods I have used to verify them. The Cabri software package has been used to indicate the correctness of each result. In certain cases the coordinate geometry involved is so complex that Derive had to be used in the proofs. One of the results I have been unable to prove, and this is therefore labelled as a conjecture, though the accuracy of Cabri, which allows figures to be dragged around, is such that we may be confident that it is true.

The Eulerian numbers are not strangers to readers of the Gazette but they are not normally associated with the subject of this article, despite the similarity of their names. This article seeks to use Eulerian numbers in generalised telescoping sums, a role that is a powerful extension of an established technique – the Euler summation formula.

In the 1930s several mathematicians, principally Alonzo Church (1903-1995), Stephen Kleene (1909-1994), Emil Post (1897-1954) and Alan Turing (1912-1954), began investigating the notion of effective calculability. (A function from natural numbers to natural numbers is effectively calculable if there is some finite rule or mechanism which will calculate the value of the function for any natural number.) Central to this activity was the notion of recursiveness. Loosely, recursion is a process of defining a function by specifying each of its values in terms of previously defined values.

This article started life as an investigation into certain aspects of the Fibonacci numbers, ‘morphed’ seamlessly into the structure of some infinite matrices and finally resolved into a general set of results that link structural aspects of Fibonacci numbers with trigonometric and hyperbolic functions. It is a surprising fact, but while I can find evidence that the link between these areas has been noted in the past, I can find no evidence that the link has been systematically developed.

The well-known simple proof that the number e is irrational consists of writing e as its exponential series, multiplying by a suitable factorial, r!, and deriving the contradiction that, if e were rational, there would be an integer lying strictly between 0 and 1. This proof is referred to as a ‘trick’ in the recent highly acclaimed book Proofs from THE BOOK [1, p. 27], where the surprising statement is made that ‘This trick … isn’t even good enough to prove that e2 is irrational’. In this article we show that there are nevertheless two similar simple proofs that e2 is irrational. The first proof is in fact well known [2, p. 12]. The second, though elementary, is less obvious and certainly not well known but, together with the idea in the first proof, it implies the even stronger property that e4 is irrational.

My first introduction to the idea of an average, at the age of about eight, left me wondering why anyone would want to add up some numbers and then divide by how many there were. A helpful illustration (not forthcoming at the time!) might be as follows:

‘Suppose three boys are going on an outing and have each brought biscuits with them. They find that Alan has 7 biscuits, Jim has 6 and Tom has 11. They decide to pool their biscuits and share them out equally. So each gets (7 + 6 + 11)/3 = 24/3 = 8 biscuits. This is the average number of biscuits brought. So, they have the same total number of biscuits as they would have had if each had brought 8.’

In this article we show how to obtain by recurrence all primitive Pythagorean triples from the single basic (4, 3, 5) triple. It is all done by repeated application of two integer unimodular transformations that leave the indefinite metric x2 + y2 - z2 invariant, together with the additional transformations (i) that change the sign of x and (ii) that change the sign of y. These alone would restrict the triples to those in which x is even, y is odd and z is positive, so we then include two further transformations (iii) that exchange x and y and (iv) that change the sign of z, thereby accounting for all primitive triples. There is a bonus in extending the method beyond the first objective in that the theory, which we outline, of the Lorentz transformations involved not only provides necessary background explanation of the first part, but it also enables us to show how to solve by recurrence Diophantine equations of the form x2 + y2 = z2 + n, where n is any integer.

One admires and applauds the enterprise of anyone who uses Gauss’s 1801 Disquisitiones arithmeticae as the starting point for mathematical exploration. I enjoyed McKeon and Sherry’s description of their journey [1] and the challenge of their conjectures. They drew attention to a class of polynomials that satisfy what they called the double angle condition ((1) below). Unfortunately, their failure to work with an appropriate definition of cyclotomic polynomials seriously handicapped their computer-aided attempt to classify double angle polynomials. Once this is remedied, a pleasant classification emerges, at least for polynomials with rational coefficients, without recourse to a computer. The main aim of this article is to present this classification. A brief final section considers McKeon and Sherry’s conjectures about irreducible double angle polynomials.

In this article we usually omit limits since, in all the cases which we consider, the integration variables are in the interval (-∞, ∞).

Few mathematics students in their first year at university would be likely to list linear algebra as part of their tool box for solving problems in the calculus class. Asked to show that

the majority, at this stage of their careers, would probably not realise that techniques which they had recently learned in linear algebra had any bearing on the problem. The computer experts in their midst might even argue that there was little merit in learning how to perform such calculations since high speed computers had eliminated the need to develop the technical skills which only a few years ago would have been regarded as essential for anyone aspiring to become a mathematician.

Pascal's triangle, the Binomial expansion and the recurrence relation between its entries are all inextricably linked. In the normal course of events, the Binomial expansion leads to Pascal's Triangle, and thence to the recurrence relation between its entries. In this article we are going to reverse this process to make it possible to explore a particular type of generalisation of such interlinked structures, by generalising the recurrence relation and then exploring the resulting generalised ‘Pascal Triangle’ and ‘Binomial expansion’. Within the spectrum of generalisations considered, we find exactly four of particular significance: those concerned with the Binomial coefficients, the Stirling numbers of both kinds, and a lesser known set of numbers – the Lah numbers. We also examine the combinatorial properties of the entries in these triangles and a prime number divisibility property that they all share. Thereby, we achieve a remarkable synthesis of these different entities.

In this paper graphs that contain unique Hamiltonian cycles are introduced. The graphs are of arbitrary size and dense in the sense that their average vertex degree is greater than half the number of vertices that make up the graph. The graphs can be used to generate challenging puzzles. The problem is particularly challenging when the graph is large and the ‘method’ of solution is unknown to the solver.

In 1632, Henry Gellibrand, then the Professor of Astronomy at Gresham College, London, arranged for the publishing of the Trigonometria Britannica (T. B.) by Adrian Vlacq in Gouda the following year: the work consisted of two Books, and sets of tables of natural sines in steps of one hundredth of a degree to 15 places, as well as tables of tangents & secants to 10 places, together with their logarithms. The explanatory Book I was the last work of Henry Briggs (1559-1631), Savilian Professor of Geometry at Oxford, and was devoted mainly to the construction of his table of sines; while Book II, written by the youthful Gellibrand on the instigation of the dying Briggs, his mentor, contained instructions and examples on the use of logarithms in solving trigonometrical problems.

One of the questions that I set my students in a recent A level revision lesson was to express (√2 - 1)6 in the form a + b√2, where a and b are integers. While they were calculating the answer, it occurred to me that, as a consequence of the fact that (√2 - 1)n → 0 as n → ∞ (since 0 < √2 - 1 < 1 ), a and b could be used to give us a ‘good’ rational approximation to √2. Within a minute or so everyone agreed that (√2 - 1)6 = 99 - 70√2, allowing us to write √2 ≈ 99/70.

In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s.

Questions about precisely when one shape fits into another attract wide attention. In 1993 Post [1] gave necessary and sufficient conditions on the six sides of two triangles for the first to fit into the second. Recently, the necessary and sufficient conditions for squares to fit in triangles [2], equilateral triangles in triangles [3], rectangles in triangles [4] and rectangles in rectangles [5] are given. In [2] Wetzel asked when a given triangle fits into a given square. In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s. For the sake of convenience, let α, β, γ denote the angles opposite sides a, b, c respectively, and we may assume without loss of generality that a ≥ b ≥ c, which implies that α ≥ β ≥ γ.

We define a Watt quadrilateral to be a quadrilateral with a pair of opposite sides of equal length. See Figure 1.

2. Linkages

The Watt linkage (Figure 2) has equal-length cranks AD and BC, A and B fixed, and coupler bar CD. It was devised by James Watt about 1784 to constrain the steam-engine piston rod, connected at E, the midpoint of CD, to approximate straight-line motion over a limited range.

This article explains the variety of patterns seen in sequences defined by for bilinear functions of the form . The initial confusion of very different patterns can be understood by generalising from the real numbers to the whole complex plane, and simplified further by projecting each number from the plane onto the surface of a sphere. The sphere is finally projected onto a Mercator map on which the iterative sequences take on the simplest possible pattern, creating a link with the problem of rhumb-line navigation. The examples given below are typical, and should first be tried as a reminder of what can happen.