Good morning and thank you for coming to listen to my Presidential Address. You have been listening to Meat Loaf’s Bat out of Hell and for those of you who are used to seeing me in various costumes I apologise for not appearing as Batman on a Batbike, but I know my limits (like .

The illustration on the front cover of the Gazette is a bit of a turkey. It comes from an old mathematics exercise book by Henry Smith who started it in 1843 and appears just before he does some examples on the single rule of three direct in vulgar fractions. I wonder how long it took him to do it, but there has surely been some implicit mathematics done in managing to get the wording arranged fairly equally around a circle! The rest of the exercise book is very neatly presented and illustrated, though without any indication that the work has been marked in a summative or formative way. Hence it would not meet today’s standards.

In the July 2011 Gazette Paseau [1] considers a population in which each birth is an independent event with probability g of being a girl and b of being a boy (b + g = 1). He supposes that, due to a preference for girls, all couples stop after producing k girls, where k is an integer greater or equal to 1. Each new birth will, even after some couples have stopped producing, still have the probability g of being a girl and b of being a boy. Paseau correctly states and shows that this will mean that the ratio

Expected number of girl births : Expected number of boy births is g : b.

So it is the same as if all couples stopped randomly. Paseau was showing that this result was a consequence of the ratio g : b always being the same for all families.

However in humans the ratio g : b is not always the same. The ratio g : b can vary significantly with many variables between couples, which influence either the ratio at conception or the subsequent implantation and survival of male or female foetuses to birth. For example, mothers whose menstrual cycles have a short follicular phase (normally days 1-14 of the 28-day cycle) tend to produce boys [2], resulting in a smaller g : b ratio. Also the timing of insemination within the menstrual cycle is associated with the gender outcome such that insemination early or late in the fertile period means that the offspring is more likely to be male; if insemination occurs in the middle it is more likely to be female [3]. It is therefore likely that changes in parental hormone concentrations during the fertile period can affect the sexes of the children [4].

1. Introduction

In the article [1], a follow-up to my article [2], Christian and Trustrum cite empirical evidence that the probability of a family giving birth to a boy or to a girl may vary from family to family. Letting bi and gi = 1 − bi respectively denote the probability of the ith family giving birth to a boy or to a girl, the suggestion is that bi may be distinct from bj for distinct i and j. As they point out, this has implications for the expected society-wide number of boys and girls. Following my discussion of the ‘stop after k girls’ policy, Christian and Trustrum introduce the related policy ‘stop after k girls or N children’. They argue that the expected number of children under this latter policy is an increasing function of bi (when 0 < k < N). The present article complements their discussion by examining this alternative stopping policy in more detail.

In this note we define and study certain subsets of the solutions of the Diophantine equations of the type

(1)

the coefficients a, b, c and d being integers, with gcd(a, b, c) = 1, and values of the unknowns X, Y and Z sought in rational numbers or integers, with X, Y, Z not all zero. Since (1) is a homogeneous equation it is elementary to pass from rational solutions to integral ones, and also in reverse. We chose to work with rational numbers because of the flexibility in computation they allow. This flexibility will enable us to prove that, under certain conditions, with three distinct solutions we can build infinite subsets of new solutions of (1), the properties of which are the main topic of this note.

As documented by L. E. Dickson [1], formulae said to give all the solutions, either of this general equation (1) or some particular cases, have appeared in the literature at least since the first half of the 17th century. Dickson himself provided the general formulae expressing all the solutions of (1) in rational numbers from an already found particular solution [2, pp. 44-45]. Similar study and formulae are part of a recent article [3, Lemma 2-1] where the solutions of (1) are applied to the study of a particular lattice. In essence these formulae prove that from any single solution of (1) we can compute all the others, which form an infinite set.

In this paper we show how to derive Vieta’s famous product of nested radicals for π [1] from the Archimedean iterative algorithm for π, [2, 3]. Only simple algebraic manipulations are needed.

The Archimedean iterative algorithm for calculating π uses a method involving the two equations:

and

In using this algorithm, we start with a circle of diameter 1. Imagine two regular polygons each with the same number of sides, circumscribed and inscribed to this circle. The larger one has perimeter a0, the smaller has perimeter b0. Since the perimeter of the circle is π we have b0 < π < a0. Now consider regular polygons with twice the number of sides that circumscribe and inscribe the circle and call their perimeters a1 and b1 respectively. These can be calculated from (1) and (2). Continuing in this way we generate the perimeters of inscribed and circumscribed regular polygons, and in each case the number of sides is twice the sides of the previous polygon. Clearly the sequence {an} approaches π from above and {bn} approaches π from below.

We consider two connected problems:

• • For a given but otherwise arbitrary triangle in the plane, to construct similar triangles which ‘meet’ this triangle.

• • To find the triangle so formed which has least area.

1. Constructing a triangle which meets another

These problems beg the question of what is meant by ‘meet’ and we now aim to make this precise:

When triangle XYZ is actually ‘in’ the triangle ABC, ‘meet’ is synonymous with the traditional ‘inscribe’ (such as in case (1) below). For ‘inscribe’ we understand that some of X, Y, Z may coincide with the vertices of ABC (such as case (2) below).

More generally we use ‘meet’ to extend these possibilities by allowing XYZ to meet triangle ABC with its sides produced externally (such as case (3) below).

“I know my ones. I know my twos. I know my threes… Ah, the finger man.” An unforgettable exchange between a defiant, remedial student, and a confident, driven math teacher in the movie Stand and Deliver [1]. This back-and-forth exchange presented a teachable moment for the determined teacher giving way to a class presentation of the finger manipulation for multiplying by nine, a technique familiar to most every elementary school student in the U.S. Little did I know this simple idea would lead to the heuristic thinking outlined in the sequence of topics below connecting Fermat and finger multiplication.

The problem of finding a closed-form evaluation of baffled the pioneers of calculus such as Leibniz and James Bernoulli and, following the latter’s promulgation of the problem, it became known as the Basel problem after his home town (which was also Euler’s birthplace). Euler’s early sensational success in solving the Basel problem by identifying is extremely well-documented. In this paper, we give the full details of his subsequent derivation of the general formula

(1)

where (Bn) is a sequence of ‘strange constants’. Euler’s polished account of his discovery, in which he popularised the designation of the strange constants as ‘Bernoulli numbers’, appears in Chapter 5 of Volume 2 of his great textbook Institutiones calculi differentialis [1; E212]: see [2] for an online English translation. Here, we will focus on his initial step-by-step account which appeared in his paper with Eneström number E130, written c1739, carrying the rather nondescript title De seriebus quibusdam considerationes, ‘Considerations about certain series’. (For convenience, we will just use ‘Eneström numbers’ when referencing Euler’s work: all are readily available on-line at [1].) Euler’s proof is notable for its early, sophisticated and incisive use of generating functions and for his brilliant insight that the sequence (Bn) occurring in the coefficients of the general ζ(2n) formula (1) also occurs in the Euler-Maclaurin summation formula and in the Maclaurin expansion of . By retracing Euler’s original path, we shall not only be able to admire the master in full creative flow, but also appreciate the role played by recurrence relations such as

(2)

which, as our ample list of references (which will be reviewed later) suggests, have been rediscovered over and over again in the literature. Moreover, our historical approach makes it clear that, while deriving (2) is relatively straightforward (and may be used to calculate ζ(2n) recursively as a rational multiple of π2n), it is establishing the connection between ζ(2n) and the Bernoulli numbers that was for Euler the more difficult step. Even today, this step presents pedagogical challenges depending on one’s starting definition for the Bernoulli numbers and what identities satisfied by them one is prepared to assume or derive.

The Royal Statistical Society and the Institute and Faculty of Actuaries have published a report, written by Roger Porkess [1], which draws urgent attention to the issue of statistical education: is it fit for purpose? Statistics is an ever-present part of real life, and there can be no doubt that for most pupils it is statistics that will be the most directly used part of their mathematical education.

The report follows an investigation into the perceived statistical needs of various subject ‘communities’, and is a follow-up to The future of statistics in our schools and colleges [2] by the same author and publishers. It is aimed at a wide readership, from policy-makers through curriculum providers and teachers (fairly low in the pecking order!) to authors and other providers of resources. There is no doubt that all such interested parties should be considering the issues covered by this study, and it should be read widely. The title of the report implies suggestions for cross-curricular work, but it is more essentially a plea to rethink the role of statistical education – a plea articulated more fully in [2]. It is the purpose of this article to further the debate.