I want to deal with one piece of applied mathematics which is a good example of the way in which—in nearly all applications of mathematics—the important steps are those of tentative model-building, and the testing of such models against reality. It is this feature that makes mechanics such an atypical case; there the model, Newton’s laws, is made for one beforehand. In the present instance the initial problem, whose partial solution leads, as usual, to other problems, can be stated very simply: in most cities the population density, σ, tends to fall off with distance, r, from the centre.

Through any point 0 in space a straight line POP1 can be drawn meeting the opposite edges AD, BC of a tetrahedron ABCD in the points P, P 1 respectively; another straight line QOQ1 can be drawn to meet DB, AC in Q, QI; and another straight line ROR1 to meet DC, AB in R, RI.

At the time of writing, second class stamps cost 12p and first class 17p. To minimise the amount of change in my pocket I have a preference for buying these stamps in ratios that cost a whole number of pounds.

This article grew out of a problem that I saw in an old textbook on sale in a second hand bookshop. It inspired a generalisation that leads to two sets of well known numbers (Bell and Stirling numbers), and then to a new result for one of them.

The natural numbers are the elements of the set N = {1,2,3,…}, the integers are the elements of the set Z = {0,± 1,±2,…} and the rational numbers are the elements of the set Q = {p/q:pεZ, qε6 N}. A real number is called algebraic if it is the root of a polynomial equation

P(x) = anxn + an−1xn−1 +…+ a1x + a0 = 0