You should never judge a book by its title - but I nearly always do. A mathematical colleague recommended a book to me Concrete Mathematics [1]. Just recently, someone else also referred me to the same book. Well a wink’s as good as a nod to a blind man and I now realise what a mistake I’d made; concrete is a conflation of ‘continuous’ and ‘discrete’. Like all good textbooks in mathematics, it distinguishes itself not just by the quality of its contents and presentation, but also by the superb collection of problems that it includes. One such problem - question 58 in Chapter 6 - inspired this article. There was a hint towards a generalised result, but there were no details. It did give a reference, [2, 3], but despite some efforts I found this unobtainable. Hence this article and my attempt to cover the same ground. The question posed in the book is concerned with the linear recurrence relations satisfied by the successive powers of Fibonacci numbers - so we start with the Fibonacci sequence itself.

Those who enjoy number patterns will no doubt have had many hours of pleasure exploring the Fibonacci sequence to various moduli, and especially in recognising the regularity with which various prime factors occur in thesequence (see [1]). Gill Hatch’s question is whether the occurrence of prime factors in generalised Fibonacci sequences is similarly predictable. Generalised Fibonacci sequences (Gn), abbreviated to GF sequences, are sequences of positive integers derived from the recurrence relation tn + 2 = tn + 1 + tn. In the case of the Fibonacci sequence (Fn), the first two terms are 1 and 1.

A vivid memory of mine is of being shown that certain 2 × 2 matrices were ‘really’ complex numbers. The correspondence between matrices of the form

and complex numbers

seemed magical and the additional remark that

was an unexpected bonus. The correspondence has been examined in various articles in the Gazette [1, 2]. That it is an isomorphism is seen by comparing

with

in the case of addition, and, in the case of multiplication, by comparing

with

In a recent article [1], Ron Shail has given a Cartesian proof of an interesting theorem due to J. A. Lester. This states that, for any triangle, the circumcentre O, the nine-point centre O9 and the two Fermat points F and Fʹ, (which are the points of concurrence of the joins of its vertices to the vertices of equilateral triangles drawn outwards/inwards on the opposite sides), are concyclic. He refers to Lester's own treatment as needing complex coordinates with computer-assisted algebra; his own proof uses an unpromising method, and results in similar problems. Contemplation of the configuration would suggest that the location of the point of intersection of FFʹ with the Euler line OO9 might lead to a simple proof. The theorem is in fact a corollary from the properties of a remarkable configuration originating with Morley [2, p. 209], and shown in Figure 1. He did not deduce Lester’s result, nor label the crucial point J in the diagram, which was drawn without that particular intersection. Also involved in this figure is a rectangular hyperbola known by the name of its describer Kiepert [3]. It is helpful to discuss all three together. What follows is a journey through country nowadays rather unfamiliar, avoiding the computerised motorway and using older tracks via complex numbers, trilinear coordinates and Euclidean methods which reveal much more than is apparent from a Cartesian treatment.

Apart from false position and double false position, another numerical method for calculating roots of equations was known to the Ancient Chinese. Chia Hsien in the eleventh century is reputed to have given an algorithm for calculating roots as well as describing Pascal’s triangle. The algorithm was mentioned again by the twelfth century scholar Liu I. The Chinese used the method to solve quadratics and cubics as early as 100 BC, but it was not until 1247 that Ch’in Kiu-Shao from South China published its extension to higher order polynomials in his work, Mathematics in nine chapters. A year later in the book, Sea-mirror of circle measurements, Li Yeh, who was from north China, took root-finding for granted. The fact that these quite independent writers published similar work suggests that finding the zeros of polynomials was well known by the middle of the thirteenth century. It was left to one of China’s greatest mathematicians, Chu Shi-kie’ (ca. 1280–1303), to give this algorithm its name fan fa, which means the method of the Celestial Element or sometimes the Celestial Unknown [1,2]. Translations and spellings of these older Chinese words do not always give the same result. This algorithm eventually became known as the Ruffini-Horner Method or more simply Horner’s method.

The cover picture shows a 22° halo seen in Preston on Sunday 28 October 2001 at 10 am. In contrast with a rainbow, the red band is on the inside rather than the outside, the colour bands are much thinner, and the arc is centred round the sun instead of being opposite to the sun. Also, it is not raining. Haloes occur on still sunny days when there is plenty of cirrus cloud. It is possible to see a full circle depending on how high the sun is in the sky. This never happens with a rainbow except when seen from an aeroplane. Sometimes the colours are particularly vivid at the 3 o’clock and 9 o’clock points. These vivid areas are called parhelia or mock suns. Sometimes the parhelia are all that can be seen.

Home to just over five million souls, Scotland is the most sparsely populated part of Britain. The people are overwhelmingly white (some 98.7%) and English speaking. Levels of deprivation vary considerably across the country as a whole. Some 20% of the school population was entitled to free school meals in 1995, though the figure was twice as high in the City of Glasgow, where life expectancy is 10 years below that of affluent parts of the south of England. In July 1997 proposals were presented for the creation of a Scottish parliament. Whilst the Westminster parliament would ‘remain sovereign’, many powers would be devolved to Edinburgh, including those relating to virtually every aspect of education. So today, the Scottish Executive Education Department (or SEED) administers Scottish Executive policy for pre-school and school education in co-operation with local authorities that are responsible for providing school education in their areas. No less than 96% of youngsters are educated in state schools. Schools associated with religious groups including the Roman Catholic Church were incorporated into the state system in the 1920s. The annual cost of running the whole education system is a little under £5 billion or some 9% of Scottish GDP [1, p. 17].

This article is a version, adapted to the written word, of an invited lecture given to the annual conference of Heads of Departments of Mathematical Sciences in universities, at the University of Birmingham, on 20 April 2001. It is offered here in the belief that an indication of the thinking behind the provision of Masterclasses for schoolchildren may also be of interest to schoolteachers themselves. I have been grateful for the very strong support which teachers have given, over ten years, to the Masterclasses which I have organised.