In May 2014, soon after the MA Conference in Nottingham, and soon after the passing of Roger Wheeler, a staunch member of the MA, who had been my teacher at school, I had the idea that my presidential address in April 2016 might be entitled ‘Inspiring Teachers’. The concept was a talk that might trace my mathematical experiences from grammar school sixth form, through my development in retirement as a contributor to masterclasses for the UK Mathematics Trust and the Royal Institution, learning from the students and from the inspiring teachers at whose masterclasses I assist, to the small understanding of Key Stage 2 Mathematics that I am gradually acquiring through an hour a week with some Year 6 students and their remarkable teachers in a local primary school.

A month or so later, when the organising committee was giving shape to Conference 2016, my words were taken as the title of the conference itself. It is deliberately ambiguous and it was gratifying to me to find this last April that its ambiguity had struck melodious chords with other speakers. My presidential address, however, was conceived as a lecture. Like all my lectures it was designed to be an oral presentation. It was not designed to be written down and published as an article. Please bear that in mind gentle reader, and judge accordingly. If you find something of value here I shall be delighted; if not, I shall not be surprised.

I recently wrote a solution to a student problem. The answers were the probabilities for four possible outcomes. As a check on the calculations, I wanted to say ‘As expected, these probabilities add up to 1.’ However, they didn't! Because of rounding, the probabilities (which I was quoting to four decimal places) only added up to 0.9999. To rescue the situation I tried rounding to 3DP or 5DP instead, but these didn't add up correctly either.

In maths lessons in secondary school, on certain occasions my teacher entertained our class with the following game. Each player starts with an empty 2 × 5 grid, as shown in Figure 1(a). In each round of the game, a die is rolled, and each player has to choose one of the empty boxes in their grid and fill it with the number rolled. For example, after 3 rounds, a player's grid may look like the one in Figure 1(b). After 10 rounds everyone's grid is filled, forming two 5-digit numbers. A player's score is the sum of her/his two 5-digit numbers. Players with the largest sum win the game. We shall call this the ‘dice-and-numbers game’. It is easy to think of other variants. Any child who is comfortable with column addition and understands how place value affects the size of numbers will develop a reasonable strategy for the game.

In a recent Note [1] Michael D. Hirschhorn presented high order algorithms for calculating numerically square roots and cube roots. In particular, he obtained the method

(1)

with , where the convergence is of tenth order:

We recall his idea in the case of an arbitrary square root with a > 0. Let p ⩾ 2 be a fixed integer. Our starting point is the relation

Choose

Here we present a simplification of one of the standard proofs that . We then look at extensions of the new approach, and add comments on the nature of the simplification (which relates to Step 1 below) and finally on the literature.

As in most proofs of the result, we shall actually prove that ; because

this is equivalent to the result sought.

We consider a simple extension of the familiar Tower of Hanoi puzzle. There are three vertical pegs lined up in a row and n discs. The discs have holes in their centres so that they can be threaded on to the pegs. Initially, all n discs are placed on the left-hand peg in non-increasing order of radius to form a tower, as in Figure 1. The object of the game is to transfer the entire tower to the right-hand peg by moving discs from peg to peg, one at a time according to the rules:

• (1) only a disc at the top of a tower may be moved;

• (2) a disc must never be placed on top of a smaller disc.

Metamathematics is the mathematical study of mathematics itself. Two of its most famous theorems were proved by Kurt Gödel in 1931. In a simplified form, Gödel's first incompleteness theorem states that no reasonable mathematical system can prove all the truths of mathematics. Gödel's second incompleteness theorem (also simplified) in turn states that no reasonable mathematical system can prove its own consistency. Another famous undecidability theorem is that the Continuum Hypothesis is neither provable nor refutable in standard set theory. Many of us logicians were first attracted to the field as students because we had heard something of these results. All research mathematicians know something of them too, and have at least a rough sense of why ‘we can't prove everything we want to prove’.

There are many families of polynomials in mathematics, and they often occur naturally in pairs. The Fibonacci polynomials and the Lucas polynomials, for example, are generated by the same recurrence relation but with different starting values, and there are many identities that link the two families [1]. The same is true for the Chebyshev polynomials of the first and second kinds, Tn (x) and Un (x) [2], respectively. There are two further polynomial families that are less well-known, the Chebyshev polynomials of the third and fourth kinds, Vn (x) and Wn (x) [3], respectively. Each of the four kinds is an example of an orthogonal polynomial family Pn (x), where for some appropriate weight function W (x), whenever n ≠ m. The families Tn (x) and Un (x) in particular are ubiquitous in their mathematical uses, in approximation theory, in differential equations, and in solving the Pell equation, to name but three. There are also many connections between Tn (x), Un (x), Vn (x) and Wn (x), some of which are explored here, and some of which we hope are new.

In a recent paper [1], the authors gave a combinatorial interpretation to sums of equally spaced binomial coefficients. Others have been interested in finding such sums, known as multisection of series. For example, Gould [2] derived interesting formulas but much of his work involved complicated manipulations of series. When the combinatorial approach can be implemented, it is neat and efficient. In this paper, we present another approach for finding equally spaced sums. We consider both infinite sums and partial finite sums based on generating functions and extracting coefficients.

While generating functions were first introduced by Abraham de Moivre at the end of seventeen century, its systematic use in combinatorial analysis was inspired by Leonhard Euler. Generating functions got a new birth in the twentieth century as a part of symbolic methods. As a central mathematical tool in discrete mathematics, generating functions are an essential part of the curriculum in the analysis of algorithms [3, 4]. They provide a bridge between discrete and continuous mathematics, as illustrated by the fact that the generating functions presented here appear as solutions to corresponding differential equations.

In this paper we will discuss the lemniscate curve and show that its arc length can be bisected and trisected using classical ruler and compasses construction. The method dates back to 1718 when Count Giulio Fagnano (1682-1766) first published these constructions [1]. Fagnano was self-educated in mathematics and treated the subject as a hobby. Euler was impressed by his work on this topic and recommended his admission to the Berlin Academy of Science. Euler then generalised Fagnano's work on integrals. In addition, Fagnano was employed to assist in reinforcing the dome of Saint Peter's which was in danger of collapse. Pope Benedict IV rewarded him for his work by publishing his mathematical papers. Fagnano achieved considerable international fame as a mathematician, and rightly so given the outstanding contributions which he made on a number of different topics.

It is well known that there are precisely five integer-sided triangles which have equal area, Δ, and perimeter, P. These triangles are called equable Heronian triangles.

A proof of this result was given by Whitworth [1]. Since Whitworth's time, much attention has been given to triangles whose areas are integer multiples of their perimeters, for example [2, 3]. However, as this paper will show, Heronian triangles with areas less than their perimeters have some mathematical interest.