As recently as ten years ago, professional mathematicians did most of their work at a desk, using paper and pencil. Today’s mathematician still sits at a desk, but facing a cathode-ray screen. The paper and pencil are still there, but a lot of the mathematician’s activities now involve use of the computer. In particular, powerful computer packages like Mathematica and Maple advertise themselves as systems for ‘doing mathematics on a computer’. This rapid transformation has changed the nature of doing mathematics in a fundamental way. The computer does not simply ‘assist’ the mathematician in doing business as usual; rather, it changes the nature of what is done. In particular, the logical structure of mathematical reasoning carried out with the aid of an interactive computer system is different from the structure of the more traditional form of mathematical reasoning. This article is a logician’s view of the ‘new maths’ of the computer age.

The following problems concern a rigid four-legged table whose feet may be considered as coplanar points, on a floor to be treated as rigid, conveniently smooth and large, but not flat.

1. (a) If the feet form a square, show that there is a position where all feet are on the floor.

2. (b) As (a), but with isosceles trapezium in place of square.

Of course (b) includes (a), but the diminished symmetry calls for a subtler argument.

1. (c) If the feet do not form a cyclic quadrilateral, show that there may be no position with all feet on the floor.

The contents of this paper will, it is hoped, be found to be of value to teachers of physics and radio principles, to their pupils, and to those engineers who prefer a full understanding of phenomena to a superficial knowledge.

In the Gazette, No. 248 (February, 1938), pp. 50-57, Mr. W J Dobbs gave a very interesting and complete descriptive account of the 18 equilateral “Morley” triangles of a given triangle. The article is especially memorable for its diagrams (1-8) and the present reader is asked to have them at hand for reference.

One of the long-standing problems of number theory, appealing to professional and recreational mathematicians alike, is the existence of Mersenne primes. These puzzling primes, for example 7, 31, 127 and 8191, are of the form 2P - 1, where p is itself a prime. The problem of their existence originated some 2400 years ago with the early Greek mathematicians' quest for the so-called perfect numbers, those like 6 and 28 which are the sum of their proper divisors. The connection was given in Euclid's Elements in 300 BC: if 2P - 1 is prime, then 2P-1(2P - 1) is a perfect number. Much later, Euler proved that all the even perfect numbers correspond to Mersenne primes. So the interest for many years has been in finding Mersenne primes. Only 39 are known, including several monsters discovered in recent years using thousands of PCs coordinated via the internet (see [1] for information on the project). Many of us would like to know if there is any way of predicting which exponents yield Mersenne primes and whether there is an infinite number of them.

The Junior Mathematical Association of London, of which I expect very few of you have even heard, is an association in a sense of the word rather different from that in which it is used in the title of the Association I am now addressing. It is the name adopted by some eight large schools (of the Public School type) in and around London for a joint Mathematical Society which has been running amongst the Sixth Form mathematical specialists in those schools for nearly five years. It is essentially an association of institutions rather than of persons, and this type of organisation will at once be seen to be necessary by any one who considers the effect of the changing personnel of Sixth Forms. It possesses therefore no strict membership roll, no subscription, and, incidentally, no harassed treasurer. All the same, it is alive, as I hope to be able to show you by the end of this short paper. It might also be recorded at this point that a somewhat similar association exists amongst some of the Classical Sixths.

In a September 2005 note in The Mathematics Teacher, Gordon [1] gave a variation on the method of completing the square that fraction-phobic students are likely to find easier, along with the comment: ‘This variation is so obvious it must have been discovered many times over, but I have never seen it in print and thought it would be useful to disseminate it more widely.’ The method Gordon [1] gave is to multiply by 4a, and then add b2, to both sides of ax2 + bx = −c to produce 4a2x2 + 4abx + b2 = b2 −4ac. The left side of this last equation can be factorised as (2ax + b)2, and now we have a pure quadratic whose solution is straightforward.