From its formation to the present time, one of the major concerns of the Committee has been the identification of a common core of mathematical knowledge that pupils with one or two A-levels in mathematics could, and should, be expected to possess. The ‘Core Syllabus in Pure Mathematics’ (published by the Association in 1968 with the support of the Schools Council) was one outcome of that concern. The first report is currently being revised, along with a companion report on Applied Mathematics (and Mathematics Applied).

The paper gives an exact evaluation of the sum of the integral parts of the terms of an arithmetical progression, with application to the checking of a table of rounded-off multiples of a constant.

The Frog Problem is a well-known exercise in problem solving that involves a lot of reasonably deep mathematics that is perhaps of more interest than learning a new integration technique. I want to try to expose this mathematics here, not that it is within the scope of all school students but so that it is recognised and possibly offered to at least the brightest pupils.

The world high jump record for men is now an amazing 2.45 m (8 ft 0½ in). This requires an athlete to raise his or her the centre of gravity through a vertical distance of about 1.3 metres, something which is quite impossible by leg thrust alone. Through experience, coaches and athletes have developed techniques which maximise performance, and this paper attempts to describe these in mathematical terms. In this article, the various factors contributing to the height of a jump are investigated with the aid of information gained from a film of a Straddle-style jump. The same principles are involved in a jump using the Fosbury Flop.

Most direct proofs of Morley’s theorem are very involved. Dick Hogg’s neat geometrical proof (Note 67.3) circumvents the problem with its slightly indirect approach, but is very pleasing, none the less. It may however not be generally known that there is a direct straightforward proof using complex numbers which is simple to those familiar with the technique involved; indeed the proof is almost inevitable, once the right start has been made.

A famous problem in euclidean geometry, associated with the names of J. Steiner (1796-1867) and D. C. Lehmus (1780-1863), is to show that a triangle is isosceles when two internal bisectors are equal. Although the converse is obvious, a direct proof of the statement is far from simple. A large number of demonstrations have been devised, some valid and some spurious, and a considerable literature exists on the subject.