By analysing the forces acting on an oar during the power stroke it is possible to estimate the mean velocity given to a racing shell. Use is made of oarlock force data, obtained experimentally by the Australian Institute of Sport, to obtain accurate predictions in the case of a racing pair. Similar conclusions would hold for boats with any number of oars. The effect of altering the angle between oar shaft and blade is investigated, leading to a suggestion for a design change to improve oar efficiency. Predictions are made of the resulting improvement in boat position in a 2000-metre race.

Broadly speaking, the development of mathematical thought has proceeded along two main channels. On the one hand mathematicians have been studying properties of objects which in some obvious and intuitive sense form what is called a continuum, while others were concerned with objects which one conceived as being discrete and well separated from each other. An example of the former type of mathematics is the differential calculus; and the theory of the distribution of the prime numbers 2, 3, 5, 7, 11, … belongs to the mathematics of the discrete. The mysteries of the epsilontic—so dear to us all: “If epsilon is sufficiently small, then delta will be as small as we please”—have no direct counterpart in the theory of the natural numbers 1, 2, 3, …. The underlying idea of the continuum seems to be that of motion, while the discrete and separable appears to be more of a static nature.

It seems likely that the teaching of statistics in schools will become more common and will be the responsibility of teachers of mathematics. Many of the pupils will not be mathematical specialists.

There are questions which have only personal answers. One such question is the one in the title; another, which is intricately bound up with it, is: Why teach mathematics? On the following pages you will find my personal answers to both questions; a selection of other people's answers to one or both are listed at the end. Much longer lists could be made, and very long ones if one included also papers in other languages than English. As you will see, my list contains papers from different parts of the world and from different times in this century; it seems that many people agree, for different reasons, that one really should teach history of mathematics, but that it has to be said again and again.

The rhombic dodecahedron has been described by Hope-Jones, Math. Gazette, Vol. XX, p. 254 (1936); it can be obtained simply by taking a cube, joining each vertex to the centre, thus obtaining six square pyramids, and placing these pyramids on the faces of a second cube. The solid thus constructed is shown in fig. 1.

Suppose that Pn is some proposition about the integer n, which we want to prove for all n≥n0 (usually n0 = 0 or 1). The form of inductive argument most commonly taught in schools is the following:

A. Simple induction. If Pn0 is true, and Pn ⇒ Pn+i for each n > n0, then P" is true for all n > n0

Many readers will be familiar with the puzzle illustrated in Fig. 1, in which the triangle ABC is isosceles and angles a, b, c are given. If a, b, c are integers when expressed in degrees then it might appear, at first sight, that so must angle θ be. But this is not the case. The reader will quickly find that θ is not accessible by simple ‘angle chasing’. By this I mean marking in the

angles obtained by making the angles in any triangle sum to 180° and by making the exterior angle of any triangle equal the sum of the interior opposites. The following question arises: Given a, b, c can angle θ be obtained by pure geometry? My conjecture is that the answer is yes in those cases when angles a, b, c and θ are all multiples of the same angle. However, I am far from proving this, and have obtained geometrical derivations only for a limited number of cases, as will be seen.

An interesting and important problem in probability theory is known as ‘Bertrand’s paradox’. Consider a chord chosen ‘at random’ inside a circle: what is the probability that the length of the chord exceeds the length of side of the inscribed equilateral triangle? Three different answers have been given to this problem p = ½, ⅓ or ¼.

This article presents interesting generalisations of three well-known results related to pedal triangles and distances, and the Simson line.

The triangle whose vertices are the feet of the perpendiculars from a point P inside a triangle ABC to each of its sides AB, BC and AC, is called a pedal triangle.

For convenience, if P, P′ are points in the Euclidean plane R2, [PP′] will denote the closed segment . Excluding the endpoints, we get the open segment ] PP′ [. Further, if Px is a half line, ]Px will stand for Px - {P}.