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.

A practical method is described which uses the result

a series which we shall denote by S8 . The approach follows the same general lines as that in a previous note (The Mathematical Gazette, Vol. LIV, October 1970, p. 261) though at a more sophisticated level. We carry out the process first and discuss error analysis, as well as the series itself, afterwards.

I like an active Saturday afternoon, so it comes about that I often end up watching sport on television. The other day it was speedway racing. I gradually became aware that I was watching 16 riders riding four at a time in a complicated arrangement of heats, the object of which was to ensure that each rider rode against every other rider exactly once. Naturally (?) there were 20 heats; but the question arises as to what numbers of riders may be arranged into a tournament, given the firm speedway requirement that four riders are to be on the track at a time, and the requirement that each meets each just once. It is also a problem to arrange the riders into heats suitably.

The reports of the London University on the General School Examinations throw an interesting light on the knowledge of candidates of the subject of approximations. The 1928-30 reports make it quite clear that the subject is badly taught and that glaring errors occur very frequently The 1929 report states that this fault is general and is not confined to particular schools. When it is considered that the University examines annually about 25,000 pupils from schools all over the country there is justification for the suggestion in the title of this paper.

The title of my address is “Mathematics for the Study of Frequency Statistics.”

By frequency statistics I mean statistics which give the result of counting. A statement as to the number of teachers receiving annual salaries in excess of a certain amount would come under the head of frequency statistics : a statement as to the total amount paid annually to these teachers would not. The sort of thing that I have mainly in mind is a table giving the numbers of boys of specified heights, the heights being taken to the nearest inch, or the numbers of girls who, having worked a paper with full marks 100, get under 5 marks, 5 and under 10, and so on. I may mention that the word “frequency” is usually applied to the actual number in a specified category; the proportion of this number to the total is called the “relative frequency.”

Marilyn vos Savant is listed in the Guinnes Book of World Records Hall of Fame as the human being with the highest IQ; she makes herself useful by answering questions in her column Ask Marilyn, which appears, for example, in Parade Magazine, a publication inserted into the Sunday editions of several American newspapers.

There are infinitely many regular polygons, but we find, on extending the idea of polygons to three dimensions, that there are only five regular polyhedra, the Platonic solids. What happens then if we try to extend this idea beyond three dimensions? It turns out that, of the five Platonic solids, just the regular tetrahedron, cube and regular octahedron have analogues in all higher dimensions, the so-called regular polytopes. Brief descriptions of these mathematical objects are to be found in [1], for example.

To the mathematician, at least in the field of elementary mathematics, it is immediately apparent what is true and what is false, and for the teacher of mathematics it is the truth, the correct answer, the right reasoning, that are all-important.

In this article, I propose to examine the other side of the picture, to consider the mistakes made by our pupils and to see how they can be turned to advantage in our work as teachers.

This article starts with a problem in probability and leads on to some natural connections with Fibonacci numbers. Details of some of the proofs are left for interested readers (or their students!).