I was prompted to write this article after two of my students came to me (independently) with what was essentially the same query. This concerned the graphical solutions of a particular scenario occurring in Game Theory. In order to state and then fully to understand their question we need to make a little foray into this area of mathematics, which appears as one of the topics in the Decision 2 module of the OCR A level (see [1], for example).

Recall that Fermat's ‘little theorem’ says that if p is prime and a is not a multiple of p, then ap − 1 ≡ 1 mod p.

This theorem gives a possible way to detect primes, or more exactly, non-primes: if for a certain a coprime to n, a− 1 is not congruent to 1 mod n, then, by the theorem, n is not prime. A lot of composite numbers can indeed be detected by this test, but there are some that evade it. Let us give ourselves some notation and terminology to discuss them.

Triangles with a 60° angle possess many interesting geometric properties. Here is a well-known example: In ΔABC with incentre I, let the rays meet the sides AC, AB at B1, C1 respectively. If ∠B = ∠C, then IB1 = IC1, trivially. Does the converse implication hold? No – the same condition holds when ∠A = 60° (irrespective of the relation between ∠B and ∠C). In this article we study some more properties possessed by triangles having a 60° angle.

In this note we start by exploring a type of solution of the equation in positive integers

for a given p, which will enable us easily to derive a class of solutions in integers of the more general equation in positive integers

for any positive integers p and n.

In another part of this note we explore some connections between the formula we find and a particular chapter in the elementary theory of numbers.

Let π (x) count the primes p ≤ x, where x is a large real number. Euclid proved that there are infinitely many primes, so that π (x) → ∞ as x → ∞; in fact his famous argument ([1: Section 2.2]) can be used to show that

There was no further progress on the problem of the distribution of primes until Euler developed various tools for the purpose; in particular he proved in 1737 [1: Theorem 427] that

The number 6174 has the well-known property that it equals the difference between the numbers formed by rearranging its digits in descending and ascending orders :

Numbers with this property are called Kaprekar constants after the Indian mathematician D. R. Kaprekar [1]. Similarly, the general process of finding the difference between numbers formed by rearranging digits in descending and ascending orders is called the Kaprekar process.

The law of cosines for any triangle with sides of length a, b and c is c2 = a2 + b2 − 2ab cos γ, where γ is the angle opposite side c. We show that this law generalises nicely to any polygon as well as any polyhedron.

The generalisation to a quadrilateral with sides a, b, c and d is

and for a five sided polygon

As a preparation for the International Mathematical Olympiad there is a common training week for the UK and the Hungarian squads. Recently I gave a talk about cross ratio there. This weapon is not one of the most frequently used mathematical techniques. When attacking a problem, we ask what are the signs that cross ratio might be of use in finding a solution? Are there typical situations which invite us to use it? In this paper we make some suggestions about answering such questions. The reader will find proofs for well-known theorems and solutions of various problems using cross ratio.

Recall that Fermat's ‘little theorem’ says that if p is prime and a is not a multiple of p, then ap − 1 ≡ 1 mod p.

This theorem gives a possible way to detect primes, or more exactly, non-primes: if for some positive a ≤ n − 1, an − 1 is not congruent to 1 mod n, then, by the theorem, n is not prime. A lot of composite numbers can indeed be detected by this test, but there are some that evade it. In other words, there are numbers n that are composite but still satisfy an − 1 ≡ 1 mod n for all a coprime to n. Such numbers might be called ‘false primes’, but in fact they are called Carmichael numbers in honour of R. D. Carmichael, who demonstrated their existence in 1912 [1] – so the year 2012 marks their centenary. (Composite numbers that satisfy the stated condition for one particular a are called a-pseudoprimes. They are the subject of a companion article [2].)

This article, rather than purporting to be any sort of serious statistical analysis of the situation, is intended in some small way to foster further debate amongst school teachers, university lecturers, examiners, curriculum designers and policy makers regarding the level of mathematical skill and knowledge possessed in certain areas by students embarking on mathematics degree programmes. The title serves merely to set the scene, and to indicate our relatively narrow focus here. However, a whole host of mathematical words or phrases could have replaced ‘trigonometry’, and it would appear that much of the mathematics associated with straightforward A level work on functions causes even fairly bright students to struggle. Surely the study of functions ought to be regarded as a fundamental element of learning mathematics, and the acquisition of understanding in this area might consequently be seen as something of fundamental importance?

It is well known that G. Tarry [1] was the first to publish a proof that the famous thirty-six officers problem posed by L. Euler [2] in 1779 has no solution but it appears to be less well known that he was the first to devise a systematic method of constructing bimagic and trimagic squares, that is, magic squares which remain magic when each entry is replaced by its square and, in the case of trimagic squares, also when each entry is replaced by its cube. Tarry's method was outlined in [3] and was explained and slightly improved upon in a book by E. Cazalas [4] published in Paris in 1934. Recently, there has been renewed interest in this topic.

Algebraic double roots are used by [1] to motivate the following limit-free definition of derivative:

‘A function f(x) has a derivative m at x = a if

for some value c.’

As we shall see later, ‘function’ in this definition will actually be restricted to real polynomials and [1] concludes

‘We have shown how an elementary algebraic principle — double roots — can lead to a complete calculus of polynomials and related functions, without the need for a limit concept.