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.

What sentence in this Mathematical Gazette contains seven a’s, four c’s, thirty-two e’s, eight f’s, five g’s, ten h’s, twelve i’s, three l’s, three m’s, fifteen n’s, eight o’s, one q, seven r’s, twenty-seven s’s, twenty-five t’s, four u’s, eight v’s, seven w’s, four y’s & two z’s?

It is a great pleasure to welcome the Mathematical Association to Cambridge, which since Newton has always been one of the most important homes of mathematics in this country. Cambridge is a traditional home of advanced and research mathematics, and it therefore gives me particular pleasure to have the honour of giving my Presidential Address to the Association in Cambridge on the topic of primary mathematics the foundation and growing point of the ‘mathematics for all’ on which our increasingly scientific and technological culture makes greater and greater demands.

In how many different ways can you pack rectangles measuring 2 cm × 1 cm into a rectangle measuring 2 cm × n cm?

I have considered elsewhere the modelling of the pricing policies of Electricity Boards, Gas Boards and British Telecom [1]. In this article the combined effect of the standing charge rebate and estimate accounts are considered and two results contrary to popular belief are shown.

Scout‘s pace, according to my old boy scout manual, consisted of alternately walking and running 22 yards. I believe it is also used by military scouts and messengers who wish to travel at a moderate speed without too much effort. As I have also discovered, it is useful for elderly, overweight athletes with a low pain tolerance.

The classical problem of the knight’s tour consists of moving a knight over a chess board in such a manner that it moves successively on to every possible square once and only once. If the initial and final squares of this tour are a knight’s move away from each other, then for obvious reasons the tour is termed re-entrant. The problem has a long and interesting history. Solutions due to De Moivre, Euler, Vandermonde, Warnsdorff and Roget, together with further references can be found in [1].

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.

Many people have had the experience of borrowing a sum of money which is to be repaid with interest by making equal monthly payments over a year or a number of years. Two problems suggest themselves:

1. If the annual rate of interest is known what is the value of each monthly payment?

2. If the monthly payment is known, what is the annual rate of interest?

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.

Now that transformations are no longer new in geometry teaching, students of mathematical education not brought up on the new orthodoxy may soon become extinct. However, I do have the good fortune to teach some members of this rare breed —mature B.Ed, students specialising in Craft, Design and Technology, and very refreshing it often is. Part of my job is to teach them about approaches to mathematics which are now common in schools, so that they know what sort of background it is reasonable to expect of their pupils.

During the recent Association Conference at Dundee some interest was expressed in the formation of mathematics workshops. Experience indicated that the main idea behind such a workshop was a place to which anyone could go with a problem in mathematics and, hopefully, receive tuition or guidance on how best to resolve it. The workshop may be attached to an existing FE college and run by the staff mainly for the benefit of students at the college but also with the possibility of helping someone from outside.

Oxbridge Tripos examination questions originating from the turn of the century have had a marked effect on the teaching of mechanics both at School and at University. These questions, reproduced in the standard textbooks of the day, were the stable diet of the undergraduate population over so many years and set the pattern for a style of question in which the student applies mechanical principles by rote without ever questioning the validity or relevance of those principles. An unfortunate consequence of this state of affairs is that mechanics has been largely overlooked as a vehicle for the teaching of mathematical modelling. There have been moves recently to make questions more “relevant” and it is not unknown to see illustrations of “real” boats, shotputters etc. However, the student is still expected to answer the question “mechanically”, if the reader will excuse the pun, without any discussion of the modelling process. How refreshing it is, even with first year undergraduates, to look at the problem of a tin can being knocked off a wall by a bullet and, instead of asking the students to apply two conservation laws in order to determine the speed with which the can hits the ground, to ask under what assumptions the two conservation laws would appropriately model the physical situation? It is quite staggering how the lecture theatre erupts with ideas, suggestions and interest.

A friend was travelling by air direct to Hong Kong. “When I leave Heathrow”, he asked, “on what bearing do you think the aeroplane will fly?” He gave me the latitudes of Heathrow and Hong Kong (approximately 51° and 22°N) and the longitude of the latter (114°E). My answer was not a good one. The reader is urged to make his or her own estimate before reading further.

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.

Currently in Northern Ireland we have two GCE Syllabuses and examinations at Advanced Level in the mathematical domain—Mathematics and Further Mathematics. In 1982 a Further Mathematics Project began work on an alternative syllabus and examination (Mode 2) for Further Mathematics. A firm proposal has now been submitted to the Northern Ireland Schools Examinations Council and has been approved by the Secondary Examinations Council.

In the traditional approach to the teaching of vectors, the scalar product of two vectors u, v is usually defined to be

|u| . |v| . cos θ

where θ is defined to be the angle between the two vectors. In a similar way, the vector product of two vectors is defined to be a vector that makes a right handed triple with u and v, and whose length is given by

|u| . |v| . sin θ

From these definitions the familiar co-ordinate expressions for dot and cross products are derived.

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.