Thomas Kuhn made his reputation by pointing out that science is not just a steady series of rational developments, but is subject to paradigm shifts – revolutions in which the entire point of view changes. A less flattering word for the process is ‘fashion’. But what is fashion other than a paradigm shift in a rather trivial area of human activity? It is often thought that mathematics is immune to paradigm shifts. After all, mathematics consists of perfect truths, proven by impeccable logic. Doesn't it? Well, no, not really – but even if it did, that wouldn't prevent paradigm shifts. Paradigm shifts are not about the internal logic; they are about the external viewpoint. They are about seeing the same facts in a new light, incorporating them into a new framework.

For a long time prime numbers have attracted the attention of mathematicians, especially those primes that possess some sort of a symmetry. The mysterious repunits An = 111… 1(n), whose decimal representations contain only units, form an important class of them. For a repunit to be prime the number n of its digits must be also prime. But this condition is far from being sufficient: for instance, A3 = 111 = 3.37 and A5 = 11111 = 41.271. Some of the repunits are nonetheless prime: A2, A19, A23, A317 and A1031, are the only known examples. The question of primeness of the repunits was discussed by M. Gardner and later in [2–4]. It is not clear whether the number of prime repunits is finite or infinite.

We all have experience of waiting in a queue. We have waited in the dentist's surgery for treatment, we have queued at the supermarket check-out with our trolley or stood in line at the Post Office to buy stamps. No doubt we have had frustrating experiences in such situations, but before we rush to criticise the discipline of a queueing system we should think about the mad stampede for the next bus that can occur in some parts of the world.

With so much changing almost daily in schools and education, it has been difficult to decide on a title for this address and even more difficult to finalize a text. Some of the titles I’ve considered are shown here.

The reader should first try the following exercise, which motivates the later work.

Exercise: Prove that the set of real matrices

forms a group under the usual matrix multiplication.

This exercise comes with a guarantee that it will provoke at least a doubletake when first encountered. The usual reaction is to protest that these matrices are obviously singular, or to complain that the identity is missing from the set. Of course those familiar with this example can feel suitably smug in their rapid dismissal of these criticisms, but what was your reaction the first time you saw it?

A student usually first meets power series through an infinite geometric progression, having previously considered finite geometric progressions. In this note we consider a variation of this introductory material which involves the Fibonacci numbers. This necessarily poses various questions, e.g. ’When does the series converge and, if so, what is the sum?’. However, there is one further intriguing question that is natural to ask, and this leads to some interesting mathematics. All of this is appropriate for sixth formers, either for classroom discussion or as an exercise.