That Spring morning, the bent old man sat down at his large oak desk with his pen momentarily poised over the familiar blue foolscap sheet. He furtively glanced around the drawing room with its old china and flowers and books in abundance and wrote in one easy movement: The principles of book-keeping by double entry. He looked up once more but this time allowed a minute to gaze out through the window to the Cam gently meandering through the meadows towards the Cambridge colleges.

As I write (Spring 2012) I am aware that I have been a member of the Mathematical Association for fifty years. I joined during the annual conference in 1962 when Dr Combridge was the main figure in the society. The highlight was a session at the conference (at King's College, London) by Geoffrey Matthews showing how matrices might be introduced into the school curriculum. Some treated this as hilarious, others as a signpost to the future. My friend Frank Budden and I regarded it as a premature ploy, which, we thought, might not tum out to be such a good idea. We subsequently wrote a successful book Mathematics through Geometry (1964) arguing in detail why the modem mathematics revolution—which was then gathering pace—might end in tears. We identified spatial imagination as the heartland of mathematics, and contrasted it with a denial of spatial imagination in the abstract calculi which were all the vogue at the time.

The celebrated theorem of Weierstrass, dating from 1885, states that continuous functions can be uniformly approximated by polynomials on any bounded, closed interval. But just how well can we approximate by polynomials of a certain degree? Let us introduce some notation to facilitate the discussion. For an interval I (which will usually be ), denote by C (I) the space of continuous functions on I, and write for (the notation is often used). Uniform convergence of fn to f (on I) equates to the statement that . Denote by the space of polynomials of degree not more than n. This is a linear subspace of C (I) of dimension n + 1. We write

Archimedes developed a method for calculating the area of a segment of a parabola. This result was outstanding in mathematical antiquity. The idea and the method were not developed further until nearly 1800 years later. In our article his method is presented in modem notation using modem concepts. We show that the method generalises to curves of higher order. Many of Archimedes' arguments can be used without any adjustments and thus show the strength of his original approach.

The mathematician from Syracuse, Archimedes (born about 287 BC, died 212 BC), developed a method for the calculation of the area of a segment of the parabola.

Apollonius was born at Perga in Pamphylia, in the south of Asia Minor. The date usually assigned to him is 240 B.C., so that he was about seventy years later than Euclid and about thirty years later than Archimedes. He tells us that it was at the suggestion of the geometer Naucrates that he wrote his treatise on Conics, consisting of eight books, and that he composed them somewhat hurriedly, and submitted them to him without revision, simply putting down the different properties as they occurred to him, intending to go over them again afterwards, the reason for his haste being the approaching departure of his friend and guest. He afterwards found time to revise and publish each book separately, and it is in this revised condition that they have come down to us.

Many readers are familiar with the celebrated Finsler-Hadwiger and Weitzenböck’s inequality in triangle geometry. On the other hand, Wolstenholme’s inequality is not widely known, but is equally important and in an indirect way can be used to derive these two. We will introduce it and then show its strength.

The Royal Statistical Society and the Institute and Faculty of Actuaries have published a report, written by Roger Porkess [1], which draws urgent attention to the issue of statistical education: is it fit for purpose? Statistics is an ever-present part of real life, and there can be no doubt that for most pupils it is statistics that will be the most directly used part of their mathematical education.

The report follows an investigation into the perceived statistical needs of various subject ‘communities’, and is a follow-up to The future of statistics in our schools and colleges [2] by the same author and publishers. It is aimed at a wide readership, from policy-makers through curriculum providers and teachers (fairly low in the pecking order!) to authors and other providers of resources. There is no doubt that all such interested parties should be considering the issues covered by this study, and it should be read widely. The title of the report implies suggestions for cross-curricular work, but it is more essentially a plea to rethink the role of statistical education – a plea articulated more fully in [2]. It is the purpose of this article to further the debate.

In [1] and [2] we demonstrated how Pascal-like triangles arose from the probabilities associated with the various outcomes of a particular game (see Definition 1 below). It was also shown that they could be considered as generalisations of Pascal's triangle. In this article we show how Fibonacci-like sequences arise from our Pascal-like triangles, and demonstrate the existence of simple relationships between these Fibonacci-like sequences and the Fibonacci sequence itself. In addition we will investigate a generalisation of the binomial coefficients that appears when considering an extended version of the game. We start by describing this game.

The Fifteen puzzle is well known. In its original form it consists of a shallow square box that can just hold sixteen small squares. Squares, numbered 1 to 15, are placed in any order in the box, leaving the sixteenth position empty (e.g. Fig. 1). The problem is to slide the squares, without lifting them out, until the numbers are in their natural order (Fig. 2).