The motion of a bowl on a grass green may be investigated mathematically, using the principles of dynamics established by Sir Isaac Newton (1642 – 1727). The analysis gives a clear picture of how the bias operates, and reveals some surprising and little-known facts which are important from the bowler's viewpoint.

John Branfield’s request [1] for the mathematics of bowls was too tempting to resist. What follows comprises a simple model for the curve upon which the bowls move, an explanation of how and why they move as they do and an attempt to refine the model of the curve. The search for a model proved to be addictive and effectively displaced many other activities. One of the reasons for this was the way in which the investigation involved so many people with interesting contributions to make. In order to give a flavour of the investigation, I have taken the liberty of giving a somewhat personal rather than strictly mathematical account, the mathematics being no less important.

George Boole, who very clearly rates as a mathematician of the very first rank, set out to solve a mathematical problem and his efforts to do so continued throughout the whole of his professional life. He never succeeded, despite the fact that he held the key to the solution for a good deal of this time. How and why he came to fail in his attempt is the subject of this paper. The difficulties encountered by students may be better understood if we realise that even the very greatest of mathematicians experience the same limitations and that they arise from the same causes.

As an elementary programming exercise I often ask students to find all three digit numbers equal to the sum of the cubes of their digits, for example 153 = 13 + 53 + 33. This is the usual solution:

Not long ago I came across the following quiz in the Sunday supplement of the Spanish newspaper El País: Check that it does not matter how you take any four numbers from the following array; provided that no two of them lie in the same row or column, their sum is always the same.

This reminded me of magic squares: square matrices where the sum of the elements in each line (row or column) and diagonal is always the same. Magic squares are a topic in recreational mathematics. Usually one looks for magic squares with elements that are integer, positive and different from each other, e.g. 1,2,…, n2 for a n × n as in

According to a Chinese legend, matrix A appeared engraved on the carapace of a turtle about 2000 BC. On the other hand, matrix B appears in Albrecht Dürer’s engraving Melencholia (1514) [1].

Polynomials like (x + l)x(x - 1) or 5 (x - 2)(x - 4)(x - 6)(x - 8) are particularly easy to sketch because their zeros are real, distinct and equally spaced along the x-axis. As a simple exercise, please picture in your mind the graph of

y=(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)

The Simson line property is normally associated with points on the circumcircle of a triangle. It is embodied by the following theorem.

Given any triangle ABC and a point P in the plane of the triangle, if perpendiculars from P on to the sides BC, CA, AB meet those sides at L, M, N respectively then L, M, N are collinear if and only if P lies on the circumcircle of triangle ABC. The line LMN is then known as the Simson line of P.

Lagrange's inversion formula is usually presented in the following form. Let f be a regular (= analytic or holomorphic) complex function with the properties

Then it is a standard theorem that it has a regular inverse function g, such that g (f (z)) = z, with similar properties. (I assume here standard results to be found in appropriate text books.

In the March 1996 issue of the Gazette, [p. 36] the following conjecture by Sir Bryan Thwaites appeared: ‘Take any set of n rational numbers. Form another set by taking the positive differences of successive members of the first set, the last such difference being formed from the last and first members of the original set. Iterate. Then in due course the set so formed will consist entirely of zeroes if and only if n is a power of two.’

1. Background

From the end of the 17th century until the 1980s, most approximations to the value of π made use of Gregory’s series:

which converges for values of tanθ between -1 and +1. Since , the simplest, and most slowly convergent, series of this kind is

Writing x = cotθ, the series becomes

Expressions for π/4 have been devised which involve the inverse cotangents of larger numbers, causing the series to converge much faster; of such expressions (referred to herein as ‘identities’) probably the most celebrated is ‘Machin’s Formula’:

This problem refers to an observation made by George Gamow and Marvin Stern [1]. They found that concerning multi-storey buildings whenever one requires the lift on a particular floor, the probability of the lift coming from above or below varies according to the height of the floor concerned. It was observed that on the second floor of a seven-storey building, the probability of the lift coming from below is very small whereas it is much greater when the sixth floor is concerned. This is very similar to the case of finding the probability of a bus coming from a particular direction at a stop on a bus route.