Diana runs from A to B with speed u and then from B to C with speed v. Eric runs from A to C with speed w. They start together and finish together.

If we now draw three lines from a point O, with OP parallel to AB and of length u, OQ parallel to BC and of length v and OR parallel to AC and of length w, then we find P, Q and R are collinear.

It is not difficult to construct a proof of the theorem so that is left as an exercise for the reader.

In this article I shall take a problem, the quadrature of the cissoid, and compare a hard (but brilliant!) pre-calculus solution of 1658 with the easy (and routine) calculus solution of only 13 years later. Afterwards we shall perhaps see the calculus through different eyes and maybe even teach it differently. Calculus is mechanical, but that it is mechanical is a marvel.

The first author discovered the property below while playing with his pocket calculator, and the second author then tried to find out what was behind this property. We enjoyed this voyage of discovery and thought that you might like to share it with us.

Actually, a calculator is not essential, but you will need a square array like in front of you. If your calculator looks different, that is no problem: all standard keyboards will work just as well.

It has been my good fortune in life to have been able to work with inspiring and generous people. When I was writing this address, I found myself thinking particularly of one, Geoffrey Matthews, to whom I still often turn when I need reminding of the funnier aspects of mathematics education. I have vivid memories of his elation, 15 years ago, when he was first approached to be President-Elect of the Mathematical Association. Being a stickler for punctuality, he went straight home and wrote his Presidential Address , more than one year in advance of its delivery. In contrast I started on this address just 10 days ago, and have had to amend it twice since, because of announcements of radical shifts in government policy. Even the title, chosen five months ago, may be out of date for reason which I will explain later.

If the above phrase appeared in an A-level paper the Board responsible would receive many protests: “a contradiction in terms”, “impossible”. But is it? Our attention was drawn to this via this A-level problem from the J.M.B.

(Further Maths, Paper II, 1988)

There are n 1 places in the world which I should like to visit. If I take the shortest possible route to visit all of these places, starting and finishing at home, what is the greatest possible distance I need to travel?

In this article we are going to look at the above problem, namely that of finding the maximin circuit length for n points (n 1 and home).

It was at the Fifth Southeast Asian Conference on Mathematical Education (June 1990, in Brunei Darussalam) that Brian Bolt posed the question: Can the numbers 1 to 12 be arranged within the hexagram below to give equal sums in the six indicated directions? This is the Magic Hexagram Problem.

MacKinnon introduced the interesting concept of an idiot-proof tile. This is a polyomino-shaped tile P which has the property that wherever two copies are placed (“by an idiot”) on the square grid, then—so long as the copies do not overlap or enclose a region (“even an idiot would not do that”)—it is possible to adjoin further copies of P so as to complete a tiling of the whole plane. MacKinnon proved that the L-triomino is an idiot-proof tile, and indicated how one could prove that the W-pentomino has the same property. The purpose of this note is to extend MacKinnon’s results, and to formulate some open problems.

For many sixth-formers and undergraduates, the method of proof-by-induction seems to resemble nothing so much as a confidence trick. They appear to have little or no appreciation of the logical foundation of the method; frequently, they have been given a mysterious recipe which might just as easily be “eye of newt and toe of frog”. They slavishly follow the rules given and so it is little wonder that they are confused when, having been cajoled incessantly not to beg the question, they are now seemingly allowed to assume the answer. However, it is not my intention in this article to present yet another justification for the principle of mathematical induction, important as that might be; in any case, perfectly good articles on the subject have already appeared in the Gazette [1,2].

People seem to regard the draw for the quarter-finals of what appears to have been the 1988 Benson & Hedges Cup in cricket as having been bungled: a quote from the relevant authority is that it was “an embarrassing cock-up”. Well, I'm not so sure! To start with, the draw as made—unlike some others!— was perfectly fair. The draw had to be restarted twice (this is what concerned observers) and so there is a question of efficiency of the chosen method of drawing. In fact, in seeking alternatives, there arise interesting questions of balancing efficiency, “excitement” and the (generally innumerate) public's perception of fairness. Let me explain!

Marilyn vos Savant is listed in the Guinnes Book of World Records Hall of Fame as the human being with the highest IQ; she makes herself useful by answering questions in her column Ask Marilyn, which appears, for example, in Parade Magazine, a publication inserted into the Sunday editions of several American newspapers.

Some time ago, I came across a book intended to popularise mathematics, whose last chapter dealt with the calculus of one variable. Its final section, evidently intended to climax the whole work, solved the problem of designing the proportions of a tin can so as to obtain the greatest volume out of a given amount of material.

This article provides solutions to three problems. The solutions are interesting in themselves; but the problems, which are very different in nature, have a common characteristic. Each can be solved relatively quickly—some would say, elegantly—by a leap of intuition. On the other hand, all three can be tackled in a more exploratory, pedestrian fashion. Some people may want to argue that one approach is “better” than another; but a more useful discussion concerns the appropriateness of a method in particular circumstances, whether it provides insight into the underlying structure and principles involved, and indeed whether this is important.

On the surface the UK SMC does not look much like a “celebration”. It consists of a one hour multiple choice paper aimed at 30-40% of 13 and 14 year olds, and is taken each year on the first Thursday in February.

Well, what would one be willing to accept as a celebration? The London Marathon has certainly captured the public imagination. Few of us can deny the emotional appeal of watching, or joining with, other human beings who are pushing themselves far beyond what is normal in everyday life. Mathematics usually evokes an equally strong emotional reaction, except that most people respond negatively. And among those few who respond positively, even fewer (in the UK) have traditionally relished the harsh glare of mathematical competition. How is it then that the UK SMC, now in only its third year, attracts a far larger entry than the London Marathon and has plans to expand to five times its present size? (The 1990 entry was roughly 42 000 from 770 schools.) Quite apart from any mathematical advantages, the scale of such an event offers youngsters a unique opportunity to participate in something truly national—a rare instance of school mathematics overcoming its instinctive introversion by becoming conscious of a challenge shared with others.

1. An automorphic number is generally understood to be an integer the final digits of whose square (and therefore of any integral power) form the original number, i.e. a is automorphic if a2 ≡ a (mod 10n), where n is the number of digits of a. There are several references to these in books on recreational mathematics, some using also bases other than 10, but otherwise generally confined to definitions and examples. Investigating some elementary properties that I had not seen treated in print led me to submit an article to the Gazette, which was published in October 1989 (no. 465, p 212). This caught the attention of Dr R. J. Cook, of Frome, who kindly sent me the results of some experiments it had led him to make on applying the idea to indices > 2. These results were surprising and at first puzzling, but a lengthy subsequent correspondence has led to the emergence of a remarkably complicated and quite fascinating picture, which this article is an attempt to summarise.

GCSE mathematics and Sixth Form Mathematics are now out of step. Sixth Form Mathematics and University Mathematics may also soon be out of step. These are two far reaching statements—but what has this got to do with rugby balls and a minibus?

By now many of us currently or recently involved in GCSE mathematics, whether as students or teachers, have become used to mathematical investigations, applied problems and project write ups all lasting about two weeks. How times have changed. It is now quite common for 15 and 16 year olds in their GCSE mathematics lesson to work in groups on open ended problems. Gone are the days (familiar to some of us!) of solving endless lists of quadratic equations and stopping without hesitation the first time b2 is less than 4ac. Generations of pupils will now, if the National Curriculum does not completely move “the goal posts”, be growing up on a new range of discoveries made possible by the introduction of fresh ideas and new starting points. “222 and all that” and “Watch out” for example are two standard pieces of coursework encountered by many pupils in their GCSE training.

The announced surprise test has long been a source of puzzlement among laymen and mathematicians alike. On the one hand, it is asserted “There will be exactly one test next week” and, on the other, it is asserted “The day of the test will be a surprise”. Simple logical analysis suggests that these two statements are in contradiction, yet, of course, when the test comes the class is surprised. By 1977, Martin Gardner had published a summary of solutions to the paradox gleaned from over 20 articles in the mathematical and philosophical literature. Most of the views expounded by Gardner admit the contradiction and implicitly invoke the logical rule that asserts that anything whatever can follow from a contradiction. This is also the approach of one of Gardner’s readers who purports to prove that the egg (the paradox of the unexpected egg, hanging, test, etc are all isomorphic) must be in box 5, yet his proof works as well for any other box! Others assume that perhaps the teacher (judge, friend with the eggs) may not be telling the truth, whereas one respondent understands "expected" in the statistical sense of “expected value”.