About the year 1700, in the course of heretofore unrecorded travels, Gulliver met islanders who used a rather odd arithmetical vocabulary They counted.

For numbers a ≥ b ≥ l we shall denote by (a, b) the set of numbers b, b + 1, …, a. We shall say that a set S of numbers is perfect if there exists a sequence containing just one pair of each of the numbers in S, satisfying the condition: for every number r in the set, the two r’s are separated by exactly r places, and having no gaps (a perfect sequence).

The problem is to arrange the numbers 1, 1, 2, 2, …, n, n in a sequence (without gaps) in such a way that for r = 1, 2, …, n the two r’s are separated by exactly r places; for example

Priday has shown in the preceding paper that for every n there exists either such a perfect sequence (as he calls it) or else a hooked sequence, with a gap one place from one end, for example

Members of the Royal Corps of Naval Constructors are responsible for the design and construction of H.M. Ships, and one of the main features of their training is the study of mathematics. The aim of this article is to give the VIth form mathematics teacher an acquaintance with the kind of mathematics used in ship design, he may then be able to encourage some of his pupils to consider the prospect of a career in the Royal Corps. Promising mathematicians with a strong practical bent can enter from school under a scheme advertised in this number of the Gazette.

This year has brought the centenary of Dirichlet’s death (5 May 1859), and it is fitting that we should recall some of his achievements and his part in the development of mathematics.

Gustav Peter Lejeune Dirichlet was born on 13 February .1805, and was the son of the postmaster at D?ren, near Cologne. Among the schools he attended Was a Gymnasium at Cologne, Where one of his teachers was the physicist Ohm. When the time came for University study in .1822, Dirichlet persuaded his parents to let him go to Paris, which Was then the world centre of mathematics. Laplace and Legendre Were still alive and active, Fourier Was at the height of his career, With a group of brilliant young men round him, and Cauchy had already begun his massive development of the theory of flmctions.

Dr. C. D. Langford recently sent me some numerical work he had done on automorphic numbers. He found automorphic numbers in the scales of 6, 10, 12, 14, and 15 and he discovered what proved to be a general method of finding an automorphic number of r + 1 digits from one of r digits. Dr. Langford’s notes stimulated me to look for explanations of his results.

Diagrammatic methods may be used to establish some well known facts about magic squares

In a magic square with 9 cells, the number in the central cell is always a, the average of the three numbers in each row, column or diagonal. Using a dot to represent the number in any cell, fill the cells of a 3 × 3 square in this way: put a dot in each of the cells forming a diagonal: these dots represent numbers totalling 3a.