In how many different ways can you pack rectangles measuring 2 cm × 1 cm into a rectangle measuring 2 cm × n cm?

The classical problem of the knight’s tour consists of moving a knight over a chess board in such a manner that it moves successively on to every possible square once and only once. If the initial and final squares of this tour are a knight’s move away from each other, then for obvious reasons the tour is termed re-entrant. The problem has a long and interesting history. Solutions due to De Moivre, Euler, Vandermonde, Warnsdorff and Roget, together with further references can be found in [1].

During the recent Association Conference at Dundee some interest was expressed in the formation of mathematics workshops. Experience indicated that the main idea behind such a workshop was a place to which anyone could go with a problem in mathematics and, hopefully, receive tuition or guidance on how best to resolve it. The workshop may be attached to an existing FE college and run by the staff mainly for the benefit of students at the college but also with the possibility of helping someone from outside.

A friend was travelling by air direct to Hong Kong. “When I leave Heathrow”, he asked, “on what bearing do you think the aeroplane will fly?” He gave me the latitudes of Heathrow and Hong Kong (approximately 51° and 22°N) and the longitude of the latter (114°E). My answer was not a good one. The reader is urged to make his or her own estimate before reading further.

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.

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.

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.