A matrix in a set M of matrices is prime (naturally enough) if it is not the product of any other matrices in the set. We thought we would look for the prime matrices in the set M of all 2 x 2 matrices with entries in the nonnegative integers and with determinant 1. To our great surprise we discovered that:

THEOREM. In M the matrices

are the only primes, and any member of M (except I) can be uniquely factorised into a product of those two.

The unit of currency on Mars is the zonk, which consists of a whole number of zinks. An enlightened, though somewhat green, government recently decided to revamp the taxation system, which at the time was manifestly unfair to male Martians as it required them to pay less tax than females. (Martians are very unselfish and feel aggrieved if they are better off than other Martians.)

Previously, only females had had to pay ‘Population Tax’, each having to pay a certain number of zonks, that number being the total number of males. (The idea was to encourage polygamous marriages.) After the change, all Martians, male and female, were to pay Population Tax, which was to be a certain number of zinks, that number being the total number of males and females.

This article was inspired by the following problem in a recent addition of the Bulletin of the IMA:

Problem. The sequence a0, a1, a2, a3,… is defined by a0 = 3 and, for n ⩾ 1, an is the real root of the equation

Prove that the sequence converges to 5.

In this short article I consider the centre of mass of a collection of particles placed on the log curve, and deduce various inequalities concerning the arithmetic, harmonic and geometric means. I have seen the idea used for a pair of points on the log curve but thought that the general approach could be of interest to Gazette readers.

The aim of this article is to present some properties of the sequence This will be shown to generate a triangle, analogous to Pascal’s and which presents interesting properties. In particular, one application consists of deriving an easy method for expressing cos nx as a polynomial in cos x.

I. Scientific background

Here are two problems in applied combinatorics.

Problem I

A plant breeder wants to assess the yields of several varieties of sugar beet. The plots of land on which the experiment is to be grown are arranged in a row: each plot is to be planted with seed of a single variety. There is enough seed for more than one appearance of each variety.

One of the difficulties encountered by students when they are first introduced to axiomatic algebra is that the axioms, such as commutativity, associativity, and distributivity, seem self-evident in the algebraic situations with which they are familiar. Indeed, the axioms are so useful, of course, because of their wide applicability. However, it can be instructive to study situations in which these axioms break down, particularly if the situations are not too contrived.