Welcome to Leicester, our first Annual Conference here since it became the home of the Association in 1975 as the result of the centenary appeal in 1971 to buy a permanent headquarters building. The city is, of course, also home of the Association's library. Welcome to the University and to the Annual Conference, one of the many manifestations in which our Association appears. It is a great pleasure and honour to be here tonight to speak on the opening day of our 1990 Conference.

The sequence 1, 3, 7, 13, 21, … arises in many ways, for example from the integers placed in a spiral on a square grid, the terms arising where the spiral intersects a diagonal on the grid, and as the central axis of a triangular array of numbers:

Is there a largest number on your calculator? There is on mine; it is 9·9999999 x 1099. It is larger than any number I have ever wanted to calculate, though it puts some known Mersenne primes out of reach. An upper boundary to the numbers with which one can calculate gives one striking difference between our imagined number line and the numbers on a calculator.

A more serious difference between the numbers on a calculator and the real number line is the existence on the calculator of a smallest number, greater than 0. On my calculator, the smallest positive number is 10-99. So, if I imagine the numbers my calculator will display marked on a number line, and zoom in on the number line to magnify it, the calculator numbers will appear spaced out, rather like integers on a conventional number line. We will see shortly that the calculator numbers are not evenly spread out, but the existence of gaps on the line, in any form, is critical. The calculator numbers are not dense on the line.

Our original source for this investigation was an Open University video on Secondary classroom practice (OU video no. PM644). Since then, however, we have seen the problem mentioned in [1], [2] and [3]. The investigation as presented to pupils is as follows:

Choose two integers between 0 and 9 (not both 0) and write them down. As an example, we shall choose 0 and 2. Write down their sum, doing the addition modulo 10. Then write down the sum of the last two numbers in the sequence, again doing the addition modulo 10. Repeat this process. Our example gives the sequence

In many situations we are faced with two numbers 0 ˂ a ≤ b as values of some variable and we are forced to produce a compromise value which in some way is intermediate between a and b. Such a compromise value can be termed an average or a mean of a and b and we would expect it to lie between a and b, with equality if and only if a = b. There are many different kinds of mean in elementary mathematics (see, for example, “Connected means” in the June 1988 Gazette). Perhaps the following three are the best known and the most important.

My younger son has become interested in the game “dungeons and dragons”. He is intrigued by the shapes of the dice that are used to play the game and by the number of faces they have. Besides the standard cubical die with its 6 faces, there are the other regular solids with 4,8,12 and 20 faces. There is also a die with 10 faces, each in the form of a kite, and my informant points out longingly that our local games shop has a die with 100 faces. “Could you have one with an odd number of faces?” he asks.

His innocent question led me to think about many mathematical aspects of dice. This article answers his question, examines the geometry of dice (highlighting some unnamed solids which deserve to be better known to model-makers), considers surprising aspects of the mechanics of tossing dice and describes sets of unusual dice whose probabilities offer scope for many games which can trap the unwary gambler.

The article describes an accessible example of dynamical chaos which has been discussed recently in the Journal of Physics. It is simple enough to be explored by microcomputer at home or school. It concerns a point “billiard ball” bouncing round an ideal square “billiard table” that rotates steadily. Without rotation the ball's motion would be a perfectly regular sequence of straight-line transits; with rotation most of the motion is unpredictably jumbled—chaotic, as we'll see. First some simple theory is needed.

When students work through a written proof what is it that they are attending to? At what level are they taking in and making sense of what they see? The mathematical symbols (i.e. diagrams, equations, etc) that are found in proofs are a result of the writer seeing some generality, attempting to express that generality and to convince themselves and others of its validity. We recognise that students, when reading a proof may work directly with the symbols, somebody else's symbols, without attempting to enter into the process of expressing generality for themselves.

We have both used the iteration x0 = 1, xn+1 = cxn (n ≥ 0) with groups of students to investigate the range of values of c > 0 for which the iterative schema converges. It is well known that there are real numbers p, q with 0 < p < q which divide the domain of positive reals into three intervals in which three distinct kinds of behaviour are exhibited. For values of c in T = (0, p) the iteration settles down to oscillations between two real values of x. For values of c in S = (p, q) the iteration converges to a single real value of x, and for values of c in D = (q, ∞) the iteration diverges. Approximations to p and q can be found by numerical investigations with calculators and computers, and the derivation of analytic expressions for p and q is within reach of many sixth-formers.