There have been many attempts to understand the portrait of the mathematician Luca Pacioli which now hangs in the Museo di Capodimonte, Naples (Plate 1), and which will be familiar to users of the Pacioli 2000 accountancy software. However none of these attempts by art historians engages with the mathematics of the picture, which the artist has gone to such trouble to detail. While to a connoisseur the painting might be thought wooden and peripheral to the history of art, I believe that the artist has recorded the most important meeting of the Renaissance.

The eminent physicist, Peter Guthrie Tait, having constructed mathematical models of the flight of a golf-ball and having carried out an extensive series of experiments to evaluate their parameters, specified the maximum distance a golf ball could be made to carry by the most accomplished golfer. He was then upstaged by his son who hit a monumental drive which pitched even further. The golfing press of the day - what there was of it - was happy to run the story and scientists themselves seemed to enjoy the joke at Tait’s expense. Macfarlane, for example, related the anecdote in the following terms in a lecture delivered just eight months after Tait’s death:

These past two years have seen the bicentenaries of Michael Faraday, Charles Babbage and John Frederick Herschel. A fourth contemporary, who deserves to rank with these, is George Green, the bicentenary of whose birth will be marked by the dedication of a plaque in Westminster Abbey. His memorial will be in proximity to those commemorating Newton, Kelvin, Faraday and Clerk Maxwell. Those to the Herschels (William and John) and Stokes are close by. Green’s memorial designates him “Mathematician and Physicist”. Most mathematicians will know of Green’s theorem and Green’s functions; physicists find his papers seminal to the study of, for example, solid state physics and elasticity and, since the mid-twentieth century, Green’s functions have become an indispensable technique for those working in nuclear physics.

Michael Deakin’s article (“A simple proof of the Beijing theorem” 76 pp251-4) will have set readers thoughts running in a number of directions. Mine turned backwards in time, towards rational approximations to the golden ratio and crystallography. The symmetries of the crystal lattice preclude the appearance of a regular pentagonal face or true 5-axis. However fair rational approximations are attained with surprising ease.

When students first meet Simpson’s rule, we look for simple applications to show them what it can do. I don’t imagine I am the only teacher who has suggested using it to find a numerical value for π - which up until then they have always had to take on trust.

Mathematica is a computer software system for symbolic manipulation and numerical computation; it has excellent graphics, a high-level, mathematically-oriented programming language and, very importantly, it takes seriously the need for work to be interactive, properly presented and communicated. This article is in fact prepared as a Mathematica “notebook”. An extensive library of supporting software “packages” has been developed, and several books about using it are now available.

There have been three obvious paradigms which have successively dominated mathematics teaching during my teaching career: three eras of predominant pedagogy. None of the so-called “reforms”, however, achieved a total take-over of the schools. “Traditional mathematics”, the first paradigm, survived right through the era of “new maths for schools”, the second paradigm. In some schools “new maths” was hardly noticed, and probably in no school did “new maths” change everything in the curriculum – there were a number of topics which carried on much as before. Applicable/problem-solving maths, the third paradigm, arrived on the back of Shell, Spode, Cockcroft and SMP 11-16 around 1982.

Consider the usual experience of a student who progresses far enough in school mathematics to begin to study the calculus. After motivation of the topic of “rates of change”, simple differentiation of polynomials is learned. More advanced functions are then considered, and eventually the student meets the product, quotient and chain rules. The result: with enough algebraic accuracy and persistence, the student can determine derived functions for virtually any sufficiently wellbehaved combination of standard functions.

Coincidences are a source of amusement and occasional unease in our lives. They often appear to defy the laws of probability – even to demand a supernatural “explanation”. Probability theory reveals the root cause of this disparity between expectation and reality. As I shall show, it stems from the fact that our intuitive estimate of the probability of a coincidence occuring becomes increasingly inaccurate the more “outlandish” the coincidence seems to be.

“There is really only one Game, the Game in which each of us is a player acting out his role. The Game is Leela, the universal play of cosmic energy.” Thus begins Harish Johari in Leela: the game of selfknowledge, a serious commentary and religious interpretation of the Hindu board game Leela. Leela is essentially the game Snakes and Ladders, which in the U.S. is the popular children’s game of snakes and ladders, England’s famous indoor sport,” for 50ø in 1943 by the Milton Bradley Company of Springfield, Massachusetts. This game is really a 101 state absorbing Markov chain, which is amenable to mathematical as well as moral analysis. In what follows we investigate only the mathematical side of this diversion, specifically the expected playing time.

A children's television game is played as follows: Four postcards from children who have written in to the programme are selected at random and the children's names are written on a board. The presenters then draw letters of the alphabet one by one from a bingo-style randomising machine and, after each letter is drawn, all occurrences of that letter in any of the children's names are crossed off. The winner of the game is the first child whose name is completely crossed off.

Any discussion of languages suitable to be taught in school mathematics classes must first attempt to answer the question as to whether computer programming ought to form any part of the mathematics curriculum. The answer is by no means obvious, especially when the view has been expressed by the editor of this journal that “all programming languages (though particularly BASIC) are inimical to mathematical thinking at a higher level ”.

Some time ago a friend, who was organising a chess league, rang me up and asked “How many different ways are there of arranging a roundrobin league tournament?” Thinking of his phone bill I told him I’d ring him back. Which was just as well, because the problem was much more difficult dian I had realised. Indeed it is unsolved in general. In the language of graph theory it is the problem of finding the number of onefactorisations of a complete graph. These and related objects have been the focus of considerable study both in combinatorics and recreational mathematics and, although the enumeration question remains unanswered in general, it is, even for small numbers of players, remarkably full of interest.