In her Gazette article on “Roll-a-penny probabilities”, Ann Hirst sets out to find the probability that a player will lose after a certain number of rolls. She considers only the simplest case where the player starts with one penny, and solves the problem by setting up a recurrence relation and using induction. The problem of what happens if the player starts with more than one penny is left for the further consideration of the reader.

The undergraduate mathematics course at Cambridge is called the Mathematical Tripos, and it has a long and distinguished history. The word Tripos refers today to a Cambridge degree course in any academic subject, which is examined by written papers. In fact, mathematics was the first Cambridge Tripos. It started in the 18th century, although its origin dates from earlier times when oral examinations were held (conducted by an examiner known as Mr Tripos, allegedly because of the three-legged stool on which he sat!). Periodically, radical changes have been made in the Mathematical Tripos, when perceived to be desirable in the light of either external events or changing emphases within the subject. A famous revolution took place in 1910, one of whose consequences was the abolition of the order of merit in which the Wranglers (candidates awarded first class honours) were placed in numerical order. The last major overhaul occurred in 1964, when the present structure was created. A similar such wholesale review has been taking place here over the last few years, with the result that a completely new course will begin in October 1991. This short article aims to describe some of the thinking behind these proposals.

The aim of this note is to discuss the relative values of different ways of posing questions on the same material. I then try to decide which methods are most effective as assessment questions. The environment of examination room, home, availability of books and notes are, of course, crucial to the selection of a method. The time available for marking must also form a realistic constraint on the choice of question.

Soon after the publication of the new edition of the SMP booklet Formulae for advanced mathematics with statistical tables, I received a letter from a colleague objecting to the statement that

“If the origin is taken at a focus of the conic,…, then the equation of the conic has the form

where e is the eccentricity …”

This article outlines possible changes to the examinations in Mathematics offered by the Scottish Examination Board as part of the Certificate of Sixth Year Studies. It must be emphasised that, at the time of writing, these changes are still only proposals. Furthermore, any views expressed in the article are the personal views of the author.

The existing CSYS courses were designed as the third stage of a progression in which pupils sat examinations at Ordinary Grade (‘O’ Grade) in S4, at Higher Grade in S5 and at CSYS Level in S6. However, the old order changeth. The Ordinary Grade is being replaced by Standard Grade and the transition will soon be complete. For articulation with Standard Grade courses, Higher Grade is being replaced by Revised Higher Grade. Finally, CSYS courses are being revised to articulate with Revised Higher. Some of the proposed revisions concern content, a good example being the situation regarding matrices. Presently pupils meet matrices in Higher but there is no mention of matrices in the formal contents of Revised Higher, so that CSYS must be adjusted accordingly. Other proposed revisions arise as a result of changes in the teaching approach. At Standard Grade and Revised Higher Grade, one of the aims is to develop pupils' skills in problem-solving. Furthermore, investigations form part of the assessment in both cases. It seems sensible that CSYS courses should continue this philosophy.

As part of a daily game-show on Channel 4 television, contestants on Countdown have to combine up to six numbers to produce a specified (but randomly chosen) three-digit result. For example, a few weeks ago they were given the numbers 4, 5, 7, 8, 9 and 10. By use of addition, subtraction, multiplication and division (provided that the result is an integer) they had to use each number at most once to obtain the number 526. And they have only 30 seconds in which to do it! One possible solution turned out to be

(7 x 9 + 4) x 8 - 10 = 526.

Roger Heath-Brown (37) is a Reader in Pure Mathematics at Oxford University, and a Tutorial Fellow at Magdalen College. He works in analytic number theory, with interests covering the distribution of prime numbers, Waring's problem and Fermat's Last Theorem

This term I'm giving a first year lecture course—Groups, Rings and Fields. I'm not at my best at 9 am and despite my efforts the undergraduates find me rather dull. The weaker ones have to have everything spelled out in full, with the result that the interesting things are too sparsely scattered for the stronger students. I'm back in college by 10.30 am having collected a pile of mail. There's a request from the Mathematical Gazette for a “day in the life of” style article, and I toy with the idea of describing an occasion on which a journal asks me to describe a typical day's work …. I also receive a referee's report on a paper submitted to me as an editorial advisor to the London Mathematical Society. The report is not encouraging and I reject the paper—a frequent enough duty, but not one I relish. The associated paperwork fills the next three-quarters of an hour. This particular manuscript is rather too similar to something published in 1963. While double-checking this in “Reviews in Number Theory” I notice an abstract of a quite unrelated paper, by a Finnish mathematician.