(This article is the result of some work done by a first-year undergraduate, Tat-Sang Fung, at the University of Hong Kong. It went way beyond what his lecturer, Man-Keung Siu, had asked for, and he was so impressed that he submitted it to the Gazette. I have edited it a little, but the ideas are entirely Tat-Sang’s.)

The following theorem about circles (which can be traced back at least as far as Archimedes) does not seem to be widely known and may be used to “pep-up” a discussion of circle theorems.

THEOREM. Let two chords of a circle meet at P and make a fixed angle θ (radians, say) with each other. Then the sum of the (opposite) arc lengths, L + L′, is independent of the position of P inside the circle (and thus, taking P at the centre, this sum equals 2rθ, where r is the radius of the circle).

In a recent session of the Royal Institution Master Classes in Cambridge, the enthusiastic participants explored a computer graphics program, designed and written by Derek and Alan Ball. The program first draws a circle, and, given an integer N, marks N equally spaced points on it, these being cyclically labelled 1,2,…,N. The operator then inputs some rule, say n ↦ 5n + 8, and the program draws, for each n, the chord from the point marked n to the point marked 5n + 8 (modulo N). Before long, the children took N to be large and flower-like envelopes of the chords started to appear (as illustrated below). Here, we show how the equations of these envelopes can be obtained in a simple way with some general parameter k, and that for positive k they are all epicycloids, that is, curves obtained by rolling a circle around a fixed circle, the simplest one being a cardioid. A similar analysis shows that for negative k they are hypocycloids (see for example E. H. Lockwood’s Book of curves, C.U.P. 1961) except when k = 1 when parallel chords occur.

Over the years we have come across many different methods of proving the sine function addition formula

and we have long felt it could be both interesting and useful to gather together in one article a list of methods to illustrate how different ideas from geometry, algebra and calculus may be used to prove it.

The Entrepreneur competition in the Sunday Times (and the Portfolio competition in the Times) have the following structure. A list of (say 10) integers a(1),a(2),…,a(10) is given in the paper (in the form of share prices) together with a target integer T. Readers have a card on which is printed a second list of (say 4) integers b(1),b(2),b(3),b(4), this second list being different for each reader.