Tilings have appeared in human activities since prehistoric times. The mathematical theory of tilings contains a rich supply of interesting and sometimes surprising facts as well as many challenging problems. A vast literature exists on the subject of tiling, and almost every imaginable variant of the question ‘How can a space be tiled by replicas of a set?’ has been discussed. Figure 1 shows a tiling of a portion of the plane where each tile is of the same size and shape and has a fractal boundary.

In this article, we discuss a combinatorial problem arising from the National Lottery. The question considered is the following: Given a works (or any other) National Lottery syndicate of say n persons, how small can n be and still allow to ensure that at least one person in the syndicate has a winning ticket? Of course we will only be able to ensure that the required ticket wins the minimum prize of £10. However, we will show that a system to achieve this end can be constructed using some remarkable combinatorics which are well-known parts of modern mathematics. In fact, the system we propose for this problem has n = 290 and at least two tickets will win £10 prizes. It will be perfectly possible that the system will result in some members of the syndicate winning a fourth, third, second or even first prize. Although the combinatorics are quite interesting, it is not possible to ‘beat’ the National Lottery using our system. We have applied our system to the actual results of the National Lottery since its inception. Our experimental evidence shows that even using our seemingly efficient scheme, payout averages only about 28% of expenditure (showing an average loss of about 72%). However, in a typical draw, our system would actually have produced 5 or so ‘winning’ tickets for the syndicate.

Over sixty years ago, a schoolmaster by the name of A. P. Rollett set a problem for a lower fifth form class which was unwittingly based upon the famous Archimedean problem of the Arbelos. This caused his pupils substantial difficulties and eventually it prompted Rollett to send a Note to the Mathematical Gazette seeking the mathematical insights of the readership regarding this question. He received very helpful responses from over twenty readers whose numbers included many notable mathematicians of the day such as E. A. Maxwell, E. H. Neville and R. S. G. Rutherford etc. Solutions were also received from two members of the clergy, (Canon) D. B. Eperson and the Rt Revd the Bishop of Kootenay, which may suggest that this problem was of sufficient intractability as to be deserving of divine inspiration! But this was not the end of it because, during the next twentythree years, there followed a series of very interesting related notes and articles continuing up to 1960 all of which made reference to Rollett’s initial homework problem of 1937.

The object of this note is to draw readers’ attention to a very new theorem in the Euclidean geometry of the triangle and to provide a straightforward Cartesian proof. This remarkable theorem, due to Lester, asserts that in any scalene triangle the two Fermat points, the nine-point centre and the circumcentre are concyclic. Lester’s original computer-assisted discovery and proof make use of her theory of ‘complex triangle coordinates’ and ‘complex triangle functions’ as expounded in, and . A proof has also been given by Trott using the advanced concept of Grobner bases in the reduction of systems of polynomial equations to ‘diagonal’ form. Trott’s work uses the computer algebra system Mathematica as an essential tool and he also provides an animation of the Lester circle as one vertex of the triangle is varied. Further information on the configuration is given in.

In the years 1603–1867, known as the Edo period, when Japan isolated itself from the western world, the country developed its own style of mathematics, especially geometry. Results and theorems of traditional Japanese mathematics, known as ‘Wasan’, were usually stated in the form of problems; these were originally displayed on wooden boards (‘sangaku’) hung in shrines and temples, but many later appeared in books, either handwritten with a brush or printed from wood blocks. (See [1, 2, 3] for more details.) Solutions to the problems were not given, but numerical answers were sometimes provided. The problems were written in a form of language called ‘Kanbun’, based on Chinese, which cannot be readily understood by modern Japanese readers even though written Japanese makes extensive use of Chinese characters.

A latin square of order n is an n × n array (or matrix) containing n distinct symbols (which are often taken to be the symbols 0, 1, …, n - 1, though any n symbols will do) arranged in such a way that all n of these symbols occur in the n cells of each row and also in the n cells of each column. Such squares occur in many guises: for example, as appropriate field layouts in the statistical design of experiments, as the addition or multiplication tables of groups, as a coding theory device and as a means of representing a finite geometry algebraically.

In a recent issue of the Gazette, S. Parameśwaran introduced S·P numbers. A positive integer n is an S·P number if it equals the sum s of its digits multiplied by the product p of its digits, i.e. n = sp. Subsequently, H. J. Godwin showed that in any number base the number of S·P numbers is finite, and A. F. Beardon and K. Robin McLean enumerated all S·P numbers in base 10 (namely, 1, 135, and 144). We generalise Beardon’s technique from base 10 to an arbitrary base to determine all S·P numbers in bases 2 through 12. Contrary to the claims of Godwin, we find 4-digit and 5-digit S·P numbers in base 11.

The world high jump record for men is now an amazing 2.45 m (8 ft 0½ in). This requires an athlete to raise his or her the centre of gravity through a vertical distance of about 1.3 metres, something which is quite impossible by leg thrust alone. Through experience, coaches and athletes have developed techniques which maximise performance, and this paper attempts to describe these in mathematical terms. In this article, the various factors contributing to the height of a jump are investigated with the aid of information gained from a film of a Straddle-style jump. The same principles are involved in a jump using the Fosbury Flop.