Snakes and ladders is an ancient Indian game of chance that offers amusement as well as a metaphor for life's many ups and downs. Games offer useful and fun ways of conveying ideas as well as solution techniques and this game has considerable mathematical tractability.

This note shows how snakes and ladders can be used to represent the ups and downs of share ownership and determine fair values of a multistage project that pays fixed dividends at uncertain completion times and has random returns

The following striking identities

are the cases n = 1,2,3,4 of a remarkable family given by G.J. Dostor [1]:

where m = n(2n + 1), and n = 1, 2, … The case m = −n is trivial. If m ≠ −n there are n + 1 squares of consecutive integers on the left and n on the right. We will treat the last term (m + n)2 on the left differently, and refer to it as a transition term relating two sums of squares of n consecutive integers.

Over two thousand years ago, Archimedes discovered a remarkable result concerning two circles drawn with reference to a configuration of three circles and a straight line. Figure 1 displays this result.

In the figure, A, B, C are three collinear points, with B between A and C; circles ω1, ω2, ω3 are drawn on AB, BC, AC as diameters, respectively; a line l is drawn through B, perpendicular to AC; a circle ω4 is inscribed in the region bounded by {ωl, ω3, l}; and a circle ω5 is inscribed in the region bounded by {ω2, ω3, l}. The Archimedean property is that ω4 and ω5 have equal radii.

‘There is a simple argument which shows that, for any positive real number x, the digit 7 appears in the decimal representation of one of the numbers x, 2x, … , 79x. I spent a wet Saturday afternoon showing that 79 can be reduced to 42, which is best possible.’

Some thirty years ago this interesting problem was given to me verbally by David Masser, who was a reader in mathematics at Nottingham University. I solved the problem, over several days, and forgot about it. While tidying up papers in retirement I found the notes on the problem, and when I mentioned this to David he too had forgotten about it until he searched through his notes. He now encourages me to publicise the problem, which is presented here with a small extension.

Without doing the morally unspeakable or the medically infeasible, can a preference for daughters rather than sons increase their relative number? If, to be more precise, the only variable over which you have control is your number of children, can you increase the ratio.

Expected value (no. of daughters) : Expected value (no. of sons) ? Naïvely, you might think so. If for example you adopt the policy ‘stop procreating as soon as a girl is born’ won't you bear more girls compared to boys than you would otherwise? No, in fact.

One day the Tortoise was discussing with mighty Achilles the behaviour of infinite series. Achilles boasted that his mental powers were so fast that he could exhaust any series summation problem by simply adding up enough terms. But Tortoise was unimpressed:

‘Is that so? What's the sum of this series, then?’ With his tail he drew a line in the sand, and wrote

which is the same as

‘I'll give you until tomorrow to find K to four significant digits!’

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.

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

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?