Mathematics is full of wonder and surprise, and I had my first big mathematical surprise at the age of 10, in 1956.

I was keen on conjuring at the time, and came across the following ‘mind-reading’ trick un a book.

Write down a 3-figure number.

Any such number will do, as long as the first and last figures differ by 2 or more.

Now reverse your number, and subtract the smaller 3-figure number from the larger.

Following a recent revival of interest in both Thomae's function and digit functions (see [1] and [2] respectively) we present here a function providing an appealing link between the two. The former, nowadays often cited in courses on real analysis, was given by Thomae in 1875; see also [3], [4] and [5]. This function, which we denote by g(x), has the following definition:

where it is so be assumed that gcd (p, q) = 1 when x is rational.

In Figure 1, triangle ABC has circumcentre O and X, Y, Z are the midpoints of the sides BC, CA, AB respectively. P is a point in the plane of ABC not lying on any of its sides. Σ is the circumcircle and S is the circle centre O passing through P.

Points on S are defined as follows. Points U, V, W are the points of intersection of AP, BP, CP with S. Points L, M, N are the reflections of P in the lines XO, YO, ZO respectively.

In [1] Niven used the integral

to give a well-known proof of the irrationality of π. Recently Zhou and Markov [2] used a recurrence relation satisfied by this integral to present an alternative proof which may be more direct than Niven's.

Niven did not cite any references in [1] and thus the origin or Hn seems rather mysterious and ingenious. However if we heed Abel's advice to ‘study the masters’, we find that Hn emerged much more naturally from the great works of Lambert [3] and Hermite [4].

Many mathematicians' experiences with distances in the settings of geometry, analysis, and topology can lead to the impression that the only worthwhile or ‘reasonable’ distance functions are metrics. We hope to convince the reader otherwise.

Recall that a metric for a set X is a function d: X × X → [0, ∞) satisfying all of the following metric axioms:

Recall that Fermat's ‘little theorem’ says that if p is prime and a is not a multiple of p, then ap − 1 ≡ 1 mod p.

This theorem gives a possible way to detect primes, or more exactly, non-primes: if for a certain a coprime to n, a− 1 is not congruent to 1 mod n, then, by the theorem, n is not prime. A lot of composite numbers can indeed be detected by this test, but there are some that evade it. Let us give ourselves some notation and terminology to discuss them.

Let π (x) count the primes p ≤ x, where x is a large real number. Euclid proved that there are infinitely many primes, so that π (x) → ∞ as x → ∞; in fact his famous argument ([1: Section 2.2]) can be used to show that

There was no further progress on the problem of the distribution of primes until Euler developed various tools for the purpose; in particular he proved in 1737 [1: Theorem 427] that

The number 6174 has the well-known property that it equals the difference between the numbers formed by rearranging its digits in descending and ascending orders :

Numbers with this property are called Kaprekar constants after the Indian mathematician D. R. Kaprekar [1]. Similarly, the general process of finding the difference between numbers formed by rearranging digits in descending and ascending orders is called the Kaprekar process.

As a preparation for the International Mathematical Olympiad there is a common training week for the UK and the Hungarian squads. Recently I gave a talk about cross ratio there. This weapon is not one of the most frequently used mathematical techniques. When attacking a problem, we ask what are the signs that cross ratio might be of use in finding a solution? Are there typical situations which invite us to use it? In this paper we make some suggestions about answering such questions. The reader will find proofs for well-known theorems and solutions of various problems using cross ratio.

It is well known that G. Tarry [1] was the first to publish a proof that the famous thirty-six officers problem posed by L. Euler [2] in 1779 has no solution but it appears to be less well known that he was the first to devise a systematic method of constructing bimagic and trimagic squares, that is, magic squares which remain magic when each entry is replaced by its square and, in the case of trimagic squares, also when each entry is replaced by its cube. Tarry's method was outlined in [3] and was explained and slightly improved upon in a book by E. Cazalas [4] published in Paris in 1934. Recently, there has been renewed interest in this topic.

Algebraic double roots are used by [1] to motivate the following limit-free definition of derivative:

‘A function f(x) has a derivative m at x = a if

for some value c.’

As we shall see later, ‘function’ in this definition will actually be restricted to real polynomials and [1] concludes

‘We have shown how an elementary algebraic principle — double roots — can lead to a complete calculus of polynomials and related functions, without the need for a limit concept.