Most boys, and many girls, are familiar with targets. All of them find geometrical probability difficult. And if they ever come across the normal probability distribution, they find the appearance of (2π)-½ in the formula surprising, and the proof somewhat artificial. While an adequate treatment of geometrical probability involves a good deal of integration, the elements of target design do not.

Many everyday events give rise to a need for quantitative thinking, thus leading to the creation of numbers and number relations in living experience. Such development of number in the normal spheres of human activity is one of the fundamental modes of human thought, and early consideration of this initial number work may be confined to verbal form. At this stage, situations requiring number thought can be discussed as they arise, and two consecutive situations often have no more in common than the use of number itself. It is this realisation which enables number-learning to take place. Increasing awareness of number and number-relations brings greater complexity, and it is then that the notation of number is needed. This notation has to be learned in relation to the ideas represented, and there are two major forms :

(i) The figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and the notion of place position to represent other numbers.

(ii) The use of a standard measure to facilitate comparisons between certain similar quantities such as length and weight.

1. Let α, β, γ, δ be parameters of four points A, B, C, D on the in-circle x2 + y2 = a2, the coordinates of A being (a cos α, a sin α) and so on. (Fig. 1).

The tangent at A is x cos α + y sin α - a = 0. The equation

The following problem was recently proposed by Professor E. H. Thompson: A rigid body is freely pivoted at a point O through which pass three fixed axes, OA, OB, OC; if the body is given one finite rotation about each of these axes, in the order stated, what conditions must be satisfied by the angles between the axes if these three rotations are to be capable of moving the body into every position consistent with O remaining fixed?

It is clearly sufficient that OA and OC coincide with a perpendicular to OB, for this is the familiar configuration. for the three Euler angles which specify rotations about two fixed intersecting perpendicular axes.

The followitlg is a useful introduction to the proof of the theorem about the product of two series.

be two convergent series of real positive terms, and let Ur → U, Vr→V as r→∞.

The problem of finding rational integral numbers which could be made the sides of right triangles, is said by Proclus to have been solved by Pythagoras, by a method which is equivalent to the use of the formula,

where m is any odd integer.

Heath (1) quotes Proclus’ statement as follows:

“The method starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it takes the square of it, subtracts unity, and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side the hypotenuse.”

It is well known that any map on a torus can be coloured with seven colours so that no two contiguous regions are of the same colour, and that fewer than seven colours will not suffice as there exist maps of seven regions of which each one abuts on to each other. In this paper all such maps of seven regions are analysed and a simple method of constructing them is given.

1. In this note we consider polynomials of a real variable x and of degree at most n,

The coefficients ak, may be complex. We are given a “mass distribution”, that is, a set of n + 1 real numbers xi, 0 ≤ i ≤ n, such that

Is a quadrilateral in which one pair of opposite sides are equal and one pair of opposite angles are equal necessarily a parallelogram? This question arose in class recently and led me to discover much that I did not know about an elementary topic. Possibly others may be interested in the results of my investigation.

The following ‘proof’, for which I am indebted to my colleague, E. H. Lockwood, answers the question in the affirmative. The subsequent paragraph contradicts this.

1. The object of this paper may be best explained by reference to the figure. ABC is a triangle with circumcentre O and orthocentre P. What other triangles share these points with ABC?

There is clearly a doubly infinite family of such triangles. If we confine attention for the moment to those possessing the same circumradius R, we shall have a singly infinite family, which we may call the family R. These also possess in common, in addition to circumcentre and orthocentre,

(i) nine-points centre U (the middle point of OP) and nine-points circle, radius ½R ;

(ii) centroid G, where (OGUP) is harmonic.