Mathematicians are fascinated by relatios and have worked our extensive theories about them. The treatments tend to be abstract and sometimes the basic ideas are lost in the abstraction. Here we investigate some common ground between mathematical relations and down-to-earth genealogy, the study of family relations.

Family relations have been studied by mathematicians, perhaps none more playfully than Thomas P. Kirkman (the nineteenth century expert in combinatorial mathematics) in his little puzzle rhyme he sent to the Educational Times [1, p.117]:

The Earth (more precisely, the ‘geoid’ thereof) is known to approximate closely to a slightly oblate spheroid whose unique axis coincides with the Earth's axis of rotation [1,2]. (By ‘spheroid’ is meant is an ellipsoid of revolution, i.e. one with two semi-axes equal; a slightly oblate one has these two semi-axes slightly longer than the unique one.) To the nearest km, the diameter of the ‘geoid’ pole-to-pole is 43 km less than the equatorial diameter of 12756 km. There is a reduction of practical significance (0.527%) in the acceleration of free fall" at sea level between the poles and the equator, and therefore in the weight of objects. Of this, 0.345% derives directly from the rotation of the Earth; the balance of 0.182% results from the purely gravitational effect of the Earth's deviation from sphericity.

In an earlier communication to the Gazette [1], the authors in effect showed, in a somewhat complicated manner, how to evaluate the integral One can show in a simpler manner, however, how to evaluate, for integers n and m, a more general integral of the form where n ≥ m, provided that if m = 1, then n is odd.

In addition, the final section to this article shows how to extend the procedure to include integrals for which m does not even have to be an integer, and also how to integrate where such an integral converges.

This essay demonstrates an application of prime numbers to the development of a calculator program that solves sudoku puzzles. Among the positive integers, the primes—numbers with exactly two divisors, the numbers themselves and 1—are central to our thinking about numbers. They give us a basis for factoring and divisibility and they contribute to the solution of many problems in the mathematical field of number theory. More important for the purposes of this paper, they provide a way of representing numbers uniquely by prime factors. For example, 6221592 = 23 × 32 × 13 × 172 × 23, any other factorisation differing only in the order of factors.

From which position on a rugby pitch should a player choose to kick in order to convert a try? The problem is to find the optimal distance from the try-line conforming to a geometrical constraint: Law 13 of the Rugby Union code stipulates, ‘after a try has been scored, the scoring team has the right to take a place kick or drop kick at goal, on a line through the place where the try was scored’.

The mathematical problem is not new and has been treated in the Gazette on several occasions [1, 2, 3], and in each case the modelling has been in two dimensions, taking a planar view of the pitch. Figure 1 represents this ‘planar model’.

This story begins in Nick Lord's workshop session at the Joined Up Mathematics conference at Keele in April 2008. He had decided to talk about Figure 1 and the results concerning it featured in [1]. He was also having an interesting e-mail exchange with Douglas Rogers, who pointed out that iterating such constructions twice produced a triangle homothetic to the original. The exchange focused on the analogous iterated Vecten configuration which John Mason recently discussed in [2]; but Nick could not see a similarly quick argument for the results in [1], which he thus set as a ‘homework problem’ for the workshop participants.

Dedicated to the memory of Russell Towle, a remarkable man who contributed so much to geometry and to other aspects of the quality of life.

We introduce an expanded notation where r + s = n, for the binomial coefficient , and then use this expanded notation to develop theorems involving 8 binomial coefficients, analogous to the Star of David Theorem, which. in its original form, involved the 6 neighbours of a given binomial coefficient in the Pascal Triangle (see Section 3), that appeared in [1,2,3,4,5,6,7,8,9].

The probability integral theorem, which states that

is on the record for having enticed numerous mathematicians to find alternative proofs for it, over a period of more than two centuries, till recent times (for a few recent references see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]). The most popular demonstration (attributed to the French mathematician Poisson; see [11], in references of [12]), the one found in almost all textbooks, relies on the double integral

(taken over the upper-right quarter of the Cartesian plane) obtained by squaring the probability integral. By resorting to polar coordinates and writing down the above integral as:

the value of the probability integral is obtained by taking the square root of the result on the right-hand side of this latter relation.

The purpose of this paper is to present some results regarding the limit is of the sequence

In the problem proposed by A. G. Ioachimescu in 1895 [1], it is asked to be shown that the sequence , defined by , for each , is convergent and its limit lies between -2 and -1.

There have been given many generalisations and other results regarding Ioachimescu's problem in the literature (see, for example, [2, problem 3.1, p. 431], [3,4], [5, Theorem 1, parts (a) and (b)], [6, problem P2, parts (i) and (ii)], [7, 8]).