This year marks the quincentenary of the death in 1492 of Piero della Francesca whose paintings have always seemed to me to have some special attraction for anyone who is mathematically inclined. I would like to celebrate this anniversary by describing a brief geometrical exploration – and some consequent speculations – prompted by looking at one of his paintings. It is not known for certain when Piero della Francesca painted his strange, enigmatic picture of the flagellation of Christ , now to be seen in the Palazzo Ducale, Urbino, which is on the cover of this edition. It is usually held to be one of his later works, painted in his late forties before that last period of his life which he seems to have devoted exclusively to mathematics. The experts disagree not only about the date of the painting, but also about its meaning and purpose. In particular, it is not known, though it is much discussed, which contemporary persons might have been portrayed as the three curiously aloof figures in the foreground.

Here is a problem which can be tackled at various levels. It has the merit of being straightforward and requiring little abstract mathematics, though it is far from trivial.

Our investigation began with the well known problem of fencing a field one of whose borders runs along a straight river. Given 100m of fencing, describe the rectangular field which has maximum area, where no fence is required along the riverbank.

In their valuable book Board games around the world. Bell and Cornelius discuss possible investigations arising from the rather complicated medieval game of Rithmomachia. One such investigation gives rise to the following problem:

Find all sets of four positive integers in which three of the four are in arithmetic progression, another three are in geometric progression, and another three are in harmonic progression (that is to say, their reciprocals are in arithmetic progression).

The authors give two such sets, and state without proof that they are the only ones, apart from multiplication throughout by a common factor. The proof that this is so is an interesting investigation in itself, and opens up some further mathematics.

The properties of linear, homogeneous, constant coefficient, second order recurrence relations such as

are well known. Consider however the recurrence relation

Such things are rarely studied; it may not be obvious that equation (2) generates a sequence which is periodic with period 9, which is to say that for any initial values x0, x1, the sequence x0, x1, x2, x3, … satisfies xn+9 = xn for every n. This article records some discoveries which resulted from an attempt to understand this interesting fact. These discoveries open up a field for exciting, elementary, but by no means trivial, investigations.

In an article by Nick MacKinnon in the December 1990 Gazette on Sophie Germain, some interesting mathematical questions were left unanswered, and I was inspired to have a go at following them up.

In correspondence with Gauss, Germain had made a conjecture which MacKinnon refined to the following statement

Conjecture. Let q be an odd prime, and a, b be coprime. If aq + bq can be represented by the form x2 + qy2, then so can a + b.

Before the last General Election politicians were arguing about the effects of a minimum wage policy. One was suggesting that we should have a minimum wage that was a certain percentage of the mean wage, while another claimed that such a policy would be inflationary, with wages spiralling up like a dog chasing its tail. What struck me was if you give some people an increase in salary to a fraction of the mean salary then you’ve increased the mean, so you’ll have to do it again. Might this process not lead to wages spiralling up to infinity?

Everyone knows that x−x = 0, but how many people realise the power of reversing this equality and writing 0 = x−x? What I mean is that we can always add and subtract the same quantity to and from any algebraic expression without altering its value. In general, we may choose infinitely many values of x, and the skill lies in choosing the most suitable x so that progress can be made in a specific problem. Beginners in mathematics are often mystified when a particular x is chosen and almost invariably respond with something like “I agree that what you are doing is correct, but why on earth did you pick that particular x?” There is no answer to this except perhaps “because it works!”.

Some three weeks into the academic year I presented, off the cuff, the following morsel to engage some lower-sixth formers who had polished off some routine derivatives all too quickly:

Sketch the curve whose equation is given by x3 + y3 + 3xy – 1 = 0 , and find the gradient at the point (–1 , –1).

Since this was near the end of a double period I thought no more of it. At the end of the next double period one student remarked that he had been having difficulty in sketching it. “It seems to be a straight line”, he said. I rashly suggested that he could well be mistaken, but promised to look at it. If my faith in the student had been greater I could have saved myself some labour searching for simple solutions of the cubic. I had noticed the solution (½ , ½) by the way. I began to have delusions of grandeur, imagining myself to be the natural successor of Diophantus, as I found the solutions (5,–4) and (–4,5).

Twelve golf balls appear identical, but eleven weigh exactly the same while one ball is either lighter or heavier than the others. Determine the “odd ball” and whether it is lighter or heavier than the others in just three weighings of the balls on a balance scale.

Puzzle problems like this abound and experience has shown that they are quite attractive to students. But this is particularly true in the present case since a solution exists that appears little short of magic. This problem provides excellent motivation for developing reasoning and communication skills. It also provides strong motivation for learning arithmetic in a base other than base 10 and for developing skill at doing mental arithmetic quickly, easily, and accurately.