The aim of this short article is to produce a sequence of real numbers x1, x2, x3, … with the property that, for each n,

x1 + x2 + … + xn = x1x2 = xn.

This turns out to be an interesting classroom exercise involving simple equations and iteration.

For many years candidates (and teachers) involved in A-level work have quoted Newton’s experimental law of restitution as one of the principles enabling collision problems to be solved. It had always seemed to me that it is fortunate that such a simple law holds and also that it is a great pity that there is no simple relationship between the initial and final kinetic energies. Recently, however, I realised that, viewed in terms of invariance, the law is not at all surprising and also that there is a very simple way of dealing with the loss of kinetic energy.

This article begins with the consideration of a short program designed to produce random values from a normal distribution and goes on to suggest how a similar program might be used to introduce confidence intervals to a group of A-level students. The programs are written in BASIC for the BBC microcomputer.

The decision by the GCE boards to ensure that all A-level Mathematics syllabuses contain a common core of pure mathematics is currently in course of implementation and near completion. In the wake of the syllabus changes a crop of new and revised A-level text books has appeared: many of these have a very familiar style and content, and have on the whole avoided any serious reappraisal of their subject matter. In particular most fail to exploit the calculator or microcomputer to any significant extent in the development of new concepts.

A very simple way of generating a sequence is to take two starting values, a1 and a2, and construct a3 as their arithmetic mean, a4 as the mean of a2 and a3 and so on. This sequence always converges to the limit and seems to be fairly widely known as a topic for investigation in schools and colleges. It can immediately be generalised to the case of k starting values, and this forms a good subject for investigation using a micro-computer. It seems likely that such a simple set-up has arisen independently many times. One emergence occurred about ten years ago at a conference of the Association of Teachers of Mathematics, where the originator introduced it under the fictitious name ‘Littov sequences’. It has recently resurfaced at a Department of Education and Science conference.

I. Scientific background

Here are two problems in applied combinatorics.

Problem I

A plant breeder wants to assess the yields of several varieties of sugar beet. The plots of land on which the experiment is to be grown are arranged in a row: each plot is to be planted with seed of a single variety. There is enough seed for more than one appearance of each variety.

The arrival of the computer will cause a number of profound changes in the calculus curriculum:

(1) Fast numerical methods will be available on the computer to give numerical solutions of problems previously handled formally by the calculus.

(2) These numerical techniques can usually be programmed in simple algorithms. The understanding of the process of the algorithm may be aided by the students carrying out their own programming.

(3) The graphic facilities of computers are providing dynamic ways of viewing the concepts, enabling them to be understood with more profound insight by students and mathematicians at all levels.

(4) Symbolic mathematical manipulators are becoming available that are able to produce the formulae for differentiating and integrating functions. This may allow more time on theoretical insight and less on specific tricks of integration.

(5) Both graphical and symbolic modes of operation are becoming available in interactive modes that enable the user to explore the concepts concerned.

(6) The methods of teaching may be modified, with exposition and exercises being enhanced by exploration, conjecture and testing by the pupils.

(7) Applications may be concerned with differential equations which may not have solutions in closed formulae. Graphic and numerical techniques used in tandem give the possibility of investigating both qualitative and quantitative aspects of the solution.

(8) The development of insights into the processes will lead to alternative approaches to the subject, e.g. numerical differentiation before symbolic differentiation, allowing the topic to be started without an initial discussion on limits. This will require a rethink of the balance between calculus theory and numerical practice and lead to modifications of the syllabus.

(9) Pressures from areas of applications such as computing, business studies, data processing, the experimental and social sciences, are demanding new techniques loosely termed “discrete mathematics”. The time for their study may reduce the time available for the calculus.

One of the difficulties encountered by students when they are first introduced to axiomatic algebra is that the axioms, such as commutativity, associativity, and distributivity, seem self-evident in the algebraic situations with which they are familiar. Indeed, the axioms are so useful, of course, because of their wide applicability. However, it can be instructive to study situations in which these axioms break down, particularly if the situations are not too contrived.

Sometimes nǃ is very divisible by n. For example, 12ǃ is divisible by 125 while 720ǃ is divisible by 720178. On the other hand, if p is prime then pǃ is divisible by p but not by p2.

In general it is quite difficult to calculate the maximum exponent m(n) such that nm(n) divides nǃ

In this short article I will locate some values of n, for which nǃ is very divisible by n, find a tight upper bound for m(n), and point the reader in the direction of related problems.

This article describes a new and general line-and-conic property, leading to a consideration of a ‘six-point’ circle and the related geometry of an interesting visual illusion.

It is an interesting exercise for a lower sixth form to investigate how muchquantitative information about the solar system can be obtained using crude apparatus, which means objects such as sticks, stones, string, coins. For the sake of saving time the concept is extended to include a ruler (which is only a graduated stick, the unit being arbitrary) and a watch (which could be replaced by keeping up continued observation of a simple pendulum).

Consider any point P inside ΔABC. The trilinear coordinates of this point are p1, p2 and p3, which are the perpendicular distances from P to the sides BC, CA and AB respectively.