This paper contains many opportunities to investigate, conjecture and verify results by writing simple computer programs, and the reader is strongly urged to participate in this way throughout.

For each positive integer n, let F(n) be the sum of the squares of the digits of n; for example,

In this article we set up a mathematical model to represent the effects of the forces which operate during the rowing of racing shells. The analysis is conducted in terms of eights, but could apply equally well to fours, pairs and double or quad sculls, and even (with obvious verbal changes) to single sculls. McMahon as well as McMahon and Bonner have previously considered various numbers of rowers in racing shells, and reached conclusions suggesting that consideration of an eight is representative of all possible combinations of rowers.

In Euclidean plane geometry there exists an interesting, although limited, duality between the concepts angle and side, similar to the general duality between points and lines in projective geometry. Perhaps surprisingly, this duality occurs quite frequently and is explored fairly extensively in.

The square is self-dual regarding these concepts as it has all angles are equal, as well as all its sides.

In this article I investigate Napoleon triangles, generalisations of the mysterious equilateral triangle in Napoleon’s theorem. I start with that theorem, develop some analogous results, find configurations with unexpected integer angles, and return to an extension of Napoleon’s theorem. Many of the geometrical proofs depend upon spiral similarities, and the numerical work uses some unfamiliar trigonometrical identities.

In 1952 W. E. Roth published two theorems, one of which has come to be known as Roth’s removal rule and (slightly generalised) goes as follows. [Recall that square matrices M, N are similar when there is an invertible matrix R such that RMR-1 = N. The matrix entries can be elements from any field, although for simplicity we shall call them ‘numbers’.]

The tide of technological advancement nearly swept away this article. I have had to rewrite this first section in the light of new developments, which I suspected might beat me to publication. For several years I have wondered why ‘pocket’ calculators, however well specified in other respects, have not had the pre-programmed ability to give values of the cumulative distribution function of a standard normal variable, commonly written as Φ (x), and its inverse. The A level mathematics student must be a prime target in the marketing of calculators and there can’t be many who don’t have to thumb through statistical tables a few hundred times in the course of their studies. The very latest generation of machines do include these functions, but it will, I suspect, be quite a few years before everyone is so well equipped.

A normal n × n magic square consists of the numbers 1, 2, … , n2 arranged in a square so that the sum of each column, row and the two diagonals is In an odd order normal magic square , the central number is This article concerns bordered magic squares of odd order, which are squares that remain magic when each outer row and column is peeled off.

Many years ago, long before it had become fashionable for so many research mathematicians to concern themselves with pedagogy, the renowned topologist Bill Thurston appeared as the keynote speaker at a symposium held at Harvard University. As he approached the podium, he could be seen carrying a set of toy building blocks, the sort designed for children. He was going to use them to construct models for his talk. But before he began, he held up the box and said, ‘You know what, kids love this stuff. What is it that we do to them?’ Of course, Thurston was right. Children have a natural curiosity, an innate sense of wonder about the world. Picasso said that children were natural born painters. Thurston said that they were natural born geometers. How do we help them to recapture that child-like sense of wonder after so many years of neglect?