The general utility of the ‘pigeon hole principle’ (that if n + 1 pigeons are placed in n boxes one of the boxes contains more than one pigeon!) is neatly illustrated by the problem of showing that:

In The bequest of the Greeks Tobias Dantzig illustrates the Euclidean division algorithm using squares and rectangles. Although he dwells on the continued fraction expansion for a/b, where a and b are relatively prime, it is only a small step to use the Greek imagery of representing numbers by geometric figures for illustrating the algorithm in finding gcd(a,b), the greatest common divisor (or highest common factor) of two numbers a and b. Indeed, this geometric line of thinking leads to an easily programmable algorithm for directly finding lcm(a,b), the least common multiple of a and b, without first finding gcd(a,b).

The change from essentially two party contests to three-, or even four-cornered fights, has provided the impetus for a fresh look at both diagrammatic representations of an election outcome, and the measure of swing from one election to the next. The representation of the votes in a three-cornered contest has been considered, amongst others, by Ibbetson [1], by Hollingdale [2] and by Miller in a series of papers and books (see for example [3] and [4]). Miller puts forward three ways of describing the outcome of such a contest, the ‘box-corner’ method (three-dimensional coordinate axes), homogeneous coordinates, and the ‘rectangular coordinate derivation’, but, as he notes, these three are fundamentally the same. Miller states that ‘mathematicians will instinctively go for the homogeneous coordinate representation’, and Hollingdale kindly obliges, although Miller himself plumps for the third representation, [3].

Given any positive real number r there will exist some positive real numbers r1, r2,…,rn all less than 4 with

That surprising ‘simultaneous factorisation’ is established in this article. It is based on some fuller work which I did a couple of years ago, but the interest in it was revived when Nick Mackinnon’s and Nick Lord’s work on ‘Sums equal to products’ was published in the March 1986 Gazette.

A persistent problem of the mathematics teacher is the motivation of students especially those not particularly interested in numerical work. In this paper we present a notational device for analysing relationships based on binary notation and show its application to the problem of incest. The authors developed this theory in the course of their study of the rabbinic law on this subject and present this application here, though it could be applied to any legal system. It is hoped that the use of this approach may be useful to introduce mathematical modes of thought to previously uninterested pupils.

Since ‘a body continues in its state of rest or of uniform motion in a straight line except in so far as it is compelled by forces to change that state’, the very first question that should be asked about walking is why any energy at all need be expended in maintaining fixed speed on level ground. With a wheeled vehicle remarkably little force is exerted for this once it is got going, and virtually all the energy is used to overcome resistive drag. But the body is not dragged, and there is little resistance presented by relaxed, healthy limbs.

Have you ever looked at a scenic photograph and wondered where it was taken from? The problem of calculating the location of the camera turns out to have a rather elegant solution. In the first place we want to make as few assumptions as possible about the optical systems used to produce the photograph. In particular, we will not, in general, know the focal length of the camera lens (or that of the enlarger). Neither will we know the precise direction in which the camera was pointing: the photograph we are looking at may be an enlargement of just a portion of the original photograph, so the centre of the photograph may not correspond to the centre of the optical system of the camera. However, we have to assume that there is no distortion, as is often evident on photographs of interiors or buildings taken with a wide angle lens. Fortunately, scenic photography usually involves bright light conditions with the camera stopped down to a small aperture so that it is a good approximation to an ideal “pin-hole” camera.

5, 11, 17, 23, 29 is an arithmetic progression of prime numbers. It is conjectured that there are such progressions with arbitrarily many terms (there cannot be infinite sequences because a, a + d, a + 2d,… sooner or later contains a + ad). The longest such known to the authors has first term 4943 and common difference 60 060 with 13 prime terms, but longer ones will have been found by now. In general, additive problems about the multiplicatively defined prime numbers are inevitably difficult (e.g. the Goldbach conjecture and twin primes) and may even be candidates for undecidability. We have been trying to prove a much more general conjecture of Erdos, spurred on by his offer of $3000 to the first proof or disproof of it.

In this article several combinatorial problems are posed for the partitioning of certain plane figures. The solutions are obtained by means of Euler’s formula for plane graphs or networks, V - E + R = 1, where V is the number of vertices, E is the number of edges, and R is the number of bounded regions of the figure. Being unified by the common thread of Euler’s formula, the sequence of problems forms an interesting unit on solving related problems.

We are just contemplating the purchase of a new car, and have been testdriving a number of competing models. A mirror fixed to the driver's door is nowadays standard equipment, and on most of the models this is attached near the front of the door, more or less in line with the hinges. But in the VW Golf it is set further back; a good feature, claimed the salesman, since it reduces the driver's “blind spot”.

I confess that it was not obvious to me that this is so. If true, then perhaps it would compensate for the inconvenience of having to turn one's head further round to look in the mirror. But mathematical investigation seemed to be called for.

Vector fields model movement over an area or a volume. For example they can be used to describe the flow of air over an aeroplane wing, or past a car as it moves, and once these models have been set up they can be adjusted to produce an efficient, aerodynamic shape.

This article is simply based on the idea of addition of vectors in the plane and its aim is to produce effective displays of vector fields on the BBC microcomputer. Using animation techniques the vector fields are represented by a flow of dots on the screen. The program has been designed so that flows can be easily combined, making it possible to manipulate them as you wish to produce your own ‘aerodynamic’ two-dimensional car designs, ship-shapes, or bridge supports.

The famous birthday problem is well known to regular Gazette readers. For those readers not familiar with it, the problem is to find the size of the smallest group of people for which there is a probability greater than one-half of two members of the group sharing a birthday. The answer (23 people) is a result which the eminent probability specialist William Feller described as “astounding”.

Serious science students and most science teachers are more or less aware of something called ‘scientific method’, and the fact that it depends on a subtle interplay involving mental constructs in the form of ‘theory’, on the basis of which one can make novel ‘predictions’ (as opposed to retrospective ‘explanations’), which can then be corroborated, or refuted, by ‘experiment’. The mental image which people have of this scientific method is often garbled, but the scientific trinity of theory, prediction, and experiment has become a commonplace. (See, for example, [17], especially pages 42–46.)

A famous unsolved problem in mathematics is the 3x + 1 conjecture. An excellent discussion of its origin and history is found in [2]. It is deceptively simple to state. Many are tempted to try it because of its enticing appearance. One aim of this note is to warn the uninitiated that a person can grow old and infirm by trying to solve the problem. There are probably fine mathematicians who have frittered away what might have been productive periods of their lives working on this devilish puzzle. The other aim is to say something at least mildly interesting about a question that I can’t answer. These are the writings of a person who has been frustrated, but not defeated, by the 3x + 1 conjecture.

Applications of convex coordinates are many and varied in mathematics today. Our first interest in convex coordinates grew out of their use in linear programming, e.g. [1]. That interest has remained quite lively as the study of convex coordinates continues to yield pleasing and sometimes unexpected results, e.g. [2,3]. Another name for these coordinates which may be more familiar to the reader is that of barycentric coordinates as originally given by the German mathematician Möbius. In this article we shall confine our attention to the Euclidean plane, but convex coordinates may be defined in Euclidean spaces of any dimension.