Suppose that a group of friends live along a straight road and that they wish to meet somewhere on that road. They wish to choose the meeting place so that they do not have to travel too far; there are several criteria which they could use:

• (1) Make the total distance travelled as small as possible.

• (2) Make the greatest distance any individual travels as small as possible.

• (3) Minimise the total of the squares of the distances travelled.

Following on from Nick MacKinnon’s work in the June 1989 Gazette concerning four different methods of square-rooting 2 × 2 matrices, we may ask if it is possible to compute all the square roots of any given 2 × 2 matrix, and if so, how? From now on by a “matrix” we mean a real or complex 2 ×2 matrix. We know that every non-zero number has precisely two square roots, but the situation is rather more complicated for matrices. In fact we shall see that the zero matrix and any other multiple of the identity matrix has an infinite number of square roots. All other matrices have only a finite number of square roots, with some (rather surprisingly) having none at all! Our method of enquiry is based on Nick MacKinnon's first method, namely matrix diagonalisation.

At one time Latin was considered such an essential part of the school curriculum that everyone was required to study it. If justification were needed for the requirement, the argument would likely be given that the study of Latin “trained the mind”. Is secondary school geometry in the same category that Latin formerly was, or are there legitimate reasons for requiring that geometry be studied by most secondary school students? Perhaps one reason for studying geometry is that it improves problem-solving ability in general.

The specific proposition that geometry improves problem-solving ability would seem to be subject to experimental verification, if indeed the study of geometry does improve problem-solving ability. The evidence herein seems to indicate that the kind of reasoning used in studying geometry improves the ability to solve not only geometric problems but other types of problems as well.

This device could be a useful teaching aid. It can easily be built by students who like to use their hands; its behaviour can easily be examined quantitatively and compared with theoretical predictions; the mathematics required to analyse its motion is not at all advanced; and it provides a good exercise in mathematical modelling. The problem appears to have been overlooked in books on mechanics, which is surprising in view of the simplicity of the device and of the mathematics associated with it.

In an article in a recent edition of Mathematics in School (May 1989), Ann Kitchen considers the mathematics of lifting bridges, among which is the bascule bridge shown below, where the counterpoise weight travels down the arc of a circle. This was chosen as a “simplification”, though even so an explicit analytical solution cannot be found. In her example the potential energy of the system was chosen to be the same in the closed and open positions, and there is then a position of unstable equilibrium in between.

A partition of n is a decomposition of n into a sum of positive integers. So the 5 partitions of 4 are

The number of partitions of n is written p(n) and (with the convection that p(0) = 1) the sequence p(0), p(1), p(2), ... begins

1, 1, 2, 3, 5, 7, 11, 15,....

Over the years there have been various articles in the Gazette [1, 2, 3] concerning the running time of cassette tapes, but none of them uses the property described here, although it is relevant to all of them.

Let us assume that the revolution counter on a cassette or video recorder is operated by the run off spool. Let a be the radius of the full spool and r the radius when an angle θ radians has been unwound. Rather as in [2] it is clear that a reasonable continuous model of this situation satisfies the equation

With the arrival of calculators having graphical displays, students at Alevel have been able to investigate the graphs of various functions where previously the analysis involved would have prevented them from doing so. Unfortunately (or fortunately?) the technology on most hand-held calculators and even on many graph-plotting packages for standard micros is not capable of handling all functions which A-level candidates might like to investigate.