There are numerous arithmetical puzzles concerning the quantities of beer that can be measured if the only available measures are a 3-pint jug and a 5-pint jug, or the exact amounts of money obtainable using only 3p and 5p coins, or the amounts that can be made with only 3p and 5p stamps.

Many elementary accounts of numerical integration use the trapezium rule and only rarely is the mid-ordinate rule mentioned. This is a great pity for the mid-ordinate rule is often simpler to apply and more accurate.

Fifteen years ago one could purchase a hand operated calculator which, by much lever pulling and handle twirling, performed the four basic arithmetical operations. Such a calculator cost around £40. Today electronic programmable calculators can be bought for prices which are less, in real terms, than that of the old mechanical calculator. As will be illustrated by examples later, such calculators incorporate the major features of a full-scale computer, and by constructing programs for a hand-held electronic calculator one can learn the basic tactics and strategies of computer programming. There are definite advantages in starting programming with a hand-held calculator; it is inexpensive, much of the ‘language’ will be familiar because hand-held calculators are now so commonly used, and one can test and use programs immediately they are written.

The practice of long jumping a motor cycle from an inclined ramp over obstacles such as parked buses has become popular with stunt-men. There have been serious accidents, some with fatal results. Some crashes have been caused by the faulty technique of keeping the throttle wide open after leaving the take-off ramp. In some cases this can lead to a backward rotation of the cycle by up to 90° during the jump, with inevitable disaster on landing.

“What is the probability that a randomly drawn triangle is acute-angled?” The problem sounds simple enough. However, what is interesting about it is the interpretation given to “random”. We investigate three possibilities

(a) that the angles of the triangle are chosen;

(b) that the lengths of the sides are chosen; and

(c) that the coordinates of the vertices of the triangle are chosen.

PASCAL is becoming increasingly available on microcomputers—the type of computer most commonly found in schools. There are strong educational arguments for teaching novices a subset of the programming language PASCAL rather than BASIC. This article presents a comparison of these two languages at an introductory level.

In this article we shall study the ‘shape’ of a triangle, and its relation to the area formula

Δ = √(s(s - a)(s - b)(s - c))

that occupied the time of sixth formers long ago. In so doing we shall resurrect a little ‘Durrel-type’ geometry, and introduce some ideas from the qualitative theory of differential equations, which are becoming increasingly familiar to undergraduates.

The purpose of this short article is to give a simple method for constructing a magic square of any given odd size. The method consists basically of two cyclic permutations perpendicular to each other.

The 5 × 5 square’s construction is illustrated, a proof outlined, and a 17 × 17 square quoted.

It follows at once from Pythagoras’ theorem about a right-angled triangle that the sum of the squares of the lengths of the diagonals of a rectangle is equal to the sum of the squares of the lengths of the four sides. Apollonius showed that the assertion holds for a parallelogram and, more recently, Amir-Moez and Hamilton gave a generalization to quadrilaterals by introducing a correction term which depends on the distance between the mid-points of the diagonals.

There is one obvious answer to this question and other less obvious answers which Twill discuss and this also gives me an opportunity to write a few words about modelling and dimensional analysis.