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.

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→∞.

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

1. The problem of forming a necklace in which no two adjacent beads have the same colour was found, unexpectedly, to lead to the little Fermat theorem and to a companion congruence with a composite modulus.

Let there be N + 1 different bead-colours. Then there are N + 1 choices of colour for the first bead to go on the string, and for each bead thereafter there are N choices. Hence for n beads the number of arrangements is (N + 1) Nn-1 But if the nth bead is the last to go on, it must have a different colour from the first.

Those who have the privilege of contributing new proofs of old theorems to the columns of the Gazette do so, normally, with complete confidence in the validity of the new proofs but possibly with some doubts as to whether the new proofs will be regarded as better than the old ones. The following attempt to set out new proofs for certain properties of the triangle differs from the norm in that there is no doubt whatever that the new proofs will be regarded as worse than the old ones and the only doubt is whether they will be regarded as valid. The doubt may arise because the aim is to deduce the truth of certain general theorems from the proved truth of one or two particular cases of the theorems. This is far from being the usual procedure in mathematics, though it is not without precedent; for example, if we show that ax2 + bx + c is zero for three values of x, it follows that it is zero for any and all values of x.