In The bequest of the Greeks Tobias Dantzig illustrates the Euclidean division algorithm using squares and rectangles. Although he dwells on the continued fraction expansion for a/b, where a and b are relatively prime, it is only a small step to use the Greek imagery of representing numbers by geometric figures for illustrating the algorithm in finding gcd(a,b), the greatest common divisor (or highest common factor) of two numbers a and b. Indeed, this geometric line of thinking leads to an easily programmable algorithm for directly finding lcm(a,b), the least common multiple of a and b, without first finding gcd(a,b).

Given any positive real number r there will exist some positive real numbers r1, r2,…,rn all less than 4 with

That surprising ‘simultaneous factorisation’ is established in this article. It is based on some fuller work which I did a couple of years ago, but the interest in it was revived when Nick Mackinnon’s and Nick Lord’s work on ‘Sums equal to products’ was published in the March 1986 Gazette.

Serious science students and most science teachers are more or less aware of something called ‘scientific method’, and the fact that it depends on a subtle interplay involving mental constructs in the form of ‘theory’, on the basis of which one can make novel ‘predictions’ (as opposed to retrospective ‘explanations’), which can then be corroborated, or refuted, by ‘experiment’. The mental image which people have of this scientific method is often garbled, but the scientific trinity of theory, prediction, and experiment has become a commonplace. (See, for example, [17], especially pages 42–46.)

A famous unsolved problem in mathematics is the 3x + 1 conjecture. An excellent discussion of its origin and history is found in [2]. It is deceptively simple to state. Many are tempted to try it because of its enticing appearance. One aim of this note is to warn the uninitiated that a person can grow old and infirm by trying to solve the problem. There are probably fine mathematicians who have frittered away what might have been productive periods of their lives working on this devilish puzzle. The other aim is to say something at least mildly interesting about a question that I can’t answer. These are the writings of a person who has been frustrated, but not defeated, by the 3x + 1 conjecture.