A latin square of order n is an n × n array (or matrix) containing n distinct symbols (which are often taken to be the symbols 0, 1, …, n - 1, though any n symbols will do) arranged in such a way that all n of these symbols occur in the n cells of each row and also in the n cells of each column. Such squares occur in many guises: for example, as appropriate field layouts in the statistical design of experiments, as the addition or multiplication tables of groups, as a coding theory device and as a means of representing a finite geometry algebraically.

David Pagni drew attention to a result which is ascribed by Dickson [2, p. 286] to Liouville (1857), that the sum of the cubes of the number of divisors of each of the divisors of an integer, is equal to the square of their sum. For example, the divisors of 6 are 1, 2, 3, and 6, which have 1, 2, 2, and 4 divisors respectively, and

Pagni observed, as have others, including Mason et al. [4, p. 179], that the reason Liouville’s result works is because the numbers generated are element-by-element products of sequences of the form {1, 2, ..., t}, which are well known to have the sum of their cubes equal to the square of their sum.

Cyclotomic polynomials, (xp - l)/(x-1), have (p-1) zeros exp(2πiq/p) (q = 1, 2, ... ,p - 1). We consider related polynomials whose zeros are given by cos(qπ/p) (q = 1, 2, ... ,p - 1) and indicate their relation to polynomials Pn (ζ) that satisfy the ‘double angle’ condition Pn(2ζ2 - l) = Pn(ζ)Pn(-ζ). The double angle condition is generalised to a chain of double angle conditions Qi(2ζ2 - 1) = Qi+1(ζ)Qi+1(-ζ), i = 1, 2, ...,n(Q1 ≡ Qn+1) For purposes of illustration, we will focus on P4(ζ), demonstrating how construction of the fourth order polynomial subject to the double angle condition yields an equation for cos . We also discuss more general features of the polynomials Pn (ζ). Furthermore, we illustrate how a simple geometrical construction can be used to derive a polynomial for cos .

In a recent issue of the Gazette, S. Parameśwaran introduced S·P numbers. A positive integer n is an S·P number if it equals the sum s of its digits multiplied by the product p of its digits, i.e. n = sp. Subsequently, H. J. Godwin showed that in any number base the number of S·P numbers is finite, and A. F. Beardon and K. Robin McLean enumerated all S·P numbers in base 10 (namely, 1, 135, and 144). We generalise Beardon’s technique from base 10 to an arbitrary base to determine all S·P numbers in bases 2 through 12. Contrary to the claims of Godwin, we find 4-digit and 5-digit S·P numbers in base 11.

The dice game ‘Jolly Roger’ may be played with 5 regular dice or 5 special ‘Jolly Roger’ dice. A ‘Jolly Roger die is a regular die in which the 1 spot is replaced by a skull and crossbones (i.e. replaced by a ‘Jolly Roger’). Play cycles through the players 5 times so that each player has 5 turns in the game. Each turn consists of 3 rolls of the 5 dice. On a particular roll of the 5 dice a player (pirate) can score points (capture cargo) only when some ‘Jolly Rogers’ appear face up. According to Koplow Games (1993):

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.