Isoperimetrical problems in ℝ2 and ℝ3 belong to the classical problems of geometry. As their name indicates one has to find geometrical bodies or areas with maximal volume, within a particular class of bodies or areas with equal surface measure. In the 19th century the isoperimetrical problem was solved for several polyhedron classes in ℝ3 [1, pp. 133-137]. Since then, a lot of research work has been devoted to more general problems where the admissable bodies were contained in the class of all compact convex subsets of ℝn or in a specific subclass, respectively. Their solutions usually can be characterized as zero sets of algebraic inequalities in surface measure and volume. The derivation of these inequalities and their solutions usually require sophisticated techniques. Of course, assertion (*) below is the expected generalisation of the well-known two-dimensional case.

A driving hazard problem was proposed by Baylis [1] and revised by Bender [2]. The problem can be described as follows: when making a right turn on British roadways, one moves to the right as far as possible on one's own side of the roadway and then turns. Unfortunately, the rear of the vehicle swings leftward as the right turn is begun - toward the unsuspecting driver passing on the left (Figure 1). This can be quite noticeable if the turning vehicle is a long bus. Hence the driving hazard problem - just how far will the back corner of the bus (P in Figure 1) swing toward the left as the bus driver negotiates the right turn - was proposed.

What a pleasure it was to read the note by Rev. L.M. Brown. It would be hard to resist exploring an article with a title like ‘An algorithm for square roots: an episode in the campaign against dotage’ under any circumstances. However, there are two reasons which made the exercise especially interesting for me.

The first reason is personal. My career as a university lecturer started in Session 1970-71 when I was appointed as a humble Demonstrator in the Technical Mathematics department at the University of Edinburgh, a group of which Dr Brown was for many years in charge. It is a delight to see him in great form at the age of 93 showing what can be done with paper, pencil and a mind still as sharp as a tack.

In the previous note, I illustrated how to construct an approximation to the factorial function x! by starting from some approximation f0 (x) on 0 ⩽ x ⩽ 1, and then extending it by the functional relation

so that

This article arose out of a first year undergraduate class project concerned with divisors. The class was presented with a sheet containing the prime factorisations and a list of divisors for the natural numbers up to 120. Among the many observations and conjectures which the students formulated some were related to the standard result about the number of divisors in terms of the prime factorisation. One conjecture made by several students which was not in the textbooks was d(n!) = 2n-1 where d(k) denotes the number of positive divisors of k. The result is true for n = 2, 3, 4, 5, which takes one to the limit of the data sheet given to the students (5! = 120). Unfortunately it breaks down for n = 6, since d(5!) = 16 but d (6!) = 30 and not 32. Several students reported that they had reached their conjecture on the basis that they thought the number of divisors of n! would double every time n increased by 1.

The offer of real money is a powerful inducement to work at a problem. So when Sir Bryan Thwaites put £100 behind his conjecture about what happens when we repeatedly take positive differences of a sequence of rational numbers, I could not resist his challenge. My solution may not have been the first to reach him, nor the most elegant, but it was fun to do. In showing that any sequence of 2n such numbers would eventually reach a string of zeros, my argument (and the one given in) depended heavily on the fact that they were rational numbers. I couldn’t help wondering what would happen if we started with a sequence of real numbers.

I just couldn't see it; that was my difficulty with many mathematical abstractions: if I couldn't visually process a maths concept I would resort to rule, hoping that some day it would make sense.

Take the Pythagorean Theorem. It has been around for two millenniums plus, readily understood by all academically exposed to the liberal arts by the time they reach puberty; but me, I knew it, a2 + b2 = c2 and the picture of three little squares huddled around the perimeter of a right triangle, that was no help. Why not three equilateral triangles around the perimeter Of course a2 √3/4 + b2 √3/4 = c2 √3/4 is a bit ungainly, but it would work.

In a previous article I wrote about the remarkable theorem of Holditch, about areas generated by a sliding or sweeping motion. While thinking further on this topic I discovered some relations concerning the areas within closed curves and the lengths of the perimeters of curves.

1. Ceva’s Theorem (Figure 1) is a familiar result in elementary geometry. It states that the product of three ratios of line segments on the sides of a triangle takes the fixed value 1. An n-gonal form of the theorem is also known, and this is just one of a large class of results concerning cyclic products of ratios of line segments on the sides or diagonals of an n-gon. In particular, suppose M is a fixed point (the Ceva point) and P = [V0, … , Vn-1] is an n-gon. Consider the lines MVi, joining the Ceva point to each of the vertices of P. These lines will cut the sides and diagonals of P in certain ratios, and each cyclic product of such ratios either takes a constant value or is related to another such cyclic product in a simple manner. The purpose of this note is to determine all such relations.