The notion of Morse code is familiar to many school pupils through interests outside of mathematics and computer studies. How can this knowledge be related to algebraic coding theory? The difficulty is that mainstream algebraic coding theory deals with block codes—in which the codewords are of uniform length. Morse code has codewords of variable length in which short codewords are used to represent common characters and longer ones for representing characters which occur less frequently (like. for E as opposed to —..—for X).

For the 3 x 3 matrix A = [a ij], the trace t of A is the sum of its main-diagonal entries:

The characteristic equation of A is

a cubic equation in λ which can be written as

where Δ is the determinant of A and

Two players A and B have a game of squash. Player A serves at the start of the game. They play a sequence of rallies. If the player serving the rally wins the rally, he scores a point, and serves the next rally. If he loses the rally, then the other player serves the next rally, with the score unaltered.

Generally the first player to reach nine points wins the game. However, if the score becomes eight-all then the first player to reach eight has a choice: he specifies whether the game is to be played to nine points or to ten ponts.

What is the correct choice? We propose to analyze the problem by use of an elementary probabilistic model.

A ball is shot at an angle of 45 degrees from the corner of a rectangular billiard table of size m × n (where m and n are integers). It always bounces off from a side at 45 degrees until it lands in a corner pocket. Will it always land in a corner or will it keep bouncing for ever? And how many times will it bounce? And which corner pocket will it land in?

Try it for yourself. For example in the 7 × 4 table shown the ball will bounce 9 times and then fall into the top left-hand corner pocket.

This is a well-known problem but we now present a simpler proof than we've seen elsewhere of the fact that the ball always ends up in a corner pocket, and we derive a formula for the number of bounces and we determine the finishing pocket.

There is at the moment considerable pressure to change the course of A-level mathematics, pressure which is now so intense that fairly radical change is both inevitable and desirable. A variety of causes have brought us to this situation. In the era of post-Cockcroft mathematics we have seen many of its recommendations implemented in the 11-16 age range. Most notable perhaps are;

1. (i) the provision of differentiated courses and examinations,

2. (ii) the encouragement of greater pupil participation in their own learning of mathematics,

3. (iii) 0 the construction of courses using a ‘bottom up’ approach.

In 1742 Goldbach sent a letter to Euler which contained the conjecture that:

Every even number greater than 2 is the sum of two prime numbers.

It is easy to verify Goldbach's conjecture numerically up to some quite high limits but no proof has yet been found. I have long had the uneasy feeling that the conjecture is not about the prime numbers at all but would equally apply to any ‘sufficiently dense’ sequence of odd numbers. However a recent attempt to generalise the sieve of Eratosthenes has led me to regard Goldbach's conjecture with a good deal more respect.