Many professional mathematicians wish to contribute to making mathematics more attractive to gifted young people. One way to pursue this goal is to create of problems that require a great deal of common sense, imagination, and, very often, a specific problem solving strategy. This article introduces such a strategy, the principle of the extreme element. Although the name is not widely used, the principle may help you to solve some non-standard mathematical problems. The material is based on my personal experience gained in extracurricular work with mathematically gifted students, and in my participation in mathematics competitions at different levels as a contestant and organiser.

Prodigiously talented with a remarkable flair for anticipating correct results and initiating fruitful areas of research, Stanislaw Ulam (‘Stan’ to his friends) was an unusual mathematician. The considerable breadth of his preoccupations, even in the halcyon days of his youth in Poland, distinguished him from his peers. Sustained by a supreme self-confidence, Ulam preferred pioneering new fields of mathematics to elaborating the ideas of others. Indeed, impatient with detail he tended to leave technicalities to those with whom he collaborated. Uprooted from his native Poland in his twenties, Ulam was to spend two thirds of his working life associated with or employed by the Los Alamos National Laboratory.

In a recent issue of the Gazette S. Parameśwaran defined a positive integer n to be an S•P number if it is the product of the sum of its digits and the product of its digits; for example, 135 and 144 are S•P numbers because

His conjecture that there are only finitely many such numbers was proved in where it was shown that any S•P number is less than 1060. Here, we prove the following result.

A positive integer, n, which is equal to the sum, S, of its digits multiplied by their product, P, is called an S·P number. S. Parameśwaran gave three examples of such numbers in:

and he conjectured that there are only a finite number of them. This was confirmed when several of us proved that each S·P number has at most 60 digits. There the matter rested for a time. Even if it were possible to test all numbers from 1 to 1060 at the rate of 1040 per second (far beyond the capacity of today’s fastest computer) it would take much longer than the estimated age of the universe to complete the task.

The genesis of this problem, which was communicated to the first author by Robert Vertes, is in a proposed ‘ice-breaking’ activity at a party attended by an even number of guests. Each guest is assigned a unique number from the set {1, 2, …, n}, where n is even, and their task is to form themselves into ½n pairs so that the sum of the numbers in each pair is a perfect square. It is not difficult to verify that this cannot be accomplished with fewer than eight people, while eight people can be split into the four pairs {1,8}, {2,7}, {3,6} and {4,5}, each summing to the square 9. Sixteen people can also be split into eight pairs with the required property, viz. the four pairs listed above together with {9, 16}, {10,15}, {11, 14} and {12, 13}. There is no requirement that every pair must sum to the same square.

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

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.

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.

In Matters for Debate, David Singmaster (DS) discusses some topics relating to the use and misuse of decimal points: the use of a full stop in writing times of the day, confusion between the decimal point and a dot used for multiplication, and writing amounts of money.