Some time last year I clipped from the Guardian a report on the bungled draw for the quarter-finals of the Benson and Hedges Cup. The relevant section is

“Perhaps we do even worse, suppressing the creative instincts of our children at the same time as researchers try to imitate them with machines. A computer is said to have applied a problem solving programme to the proposition that the base angles of an isosceles triangle are equal. Instead of the Euclidean proof which proves two right-angled-triangles [on the left below] to be congruent, the computer produced a more elegant proof from the simpler construction (sic) [on the right below]. The conventional working must have been taught to millions of children: we may reasonably ask why none of them, apparently, have discovered this simple, new proof for themselves.”

Any positive fraction can be represented by a lattice point in the positive quadrant of the coordinate plane, (x, y) being used to represent the fraction y/x. The diagram shows the points representing 2/3,4/6,6/9 and it’s easy to see that equivalent fractions will in general lie on a line through the origin, the gradient of that line equalling the value of the fractions in question.

A number of recently published articles have dealt with aspects of mathematics as applied to the motion of the human body, [1, 2, 3], some with special regard to the modelling process. There seems to be a growing interest in the application of mathematics to this area and the recent introduction of A-level “Sports Studies” into some schools and sixth form colleges must help this trend. At a talk I gave recently to a branch of the Mathematical Association on “Mechanics and gymnastics”, five out of an audience of twenty-five were P.E. specialists looking for ideas to enable them to teach the mathematical/mechanical part of this A-level syllabus.

Although it does not often feature in today’s syllabuses, the study of the incentre and circumcentre of a triangle can be a useful source of material for showing the applications of vector methods to geometry. This idea has featured in some recent Gazettes, but in this article I illustrate an alternative and essentially simpler vector approach.

Suppose you are trying to tile an infinite grid of unit squares with 2 x 1 rectangles so that each rectangle covers two squares. You place two of the rectangles without thinking. Can the tiling now be accomplished? As the answer is “yes” I call the rectangle an “idiot-proof tile”. In general a polyomino will be called idiot proof if the infinite grid can be tiled irrespective of the positions of two of the tiles (the idiotic tiles), except that the idiotic tiles shall not be overlapping and shall not completely surround an area that is less than that of the tile (not even an idiot would do that!). The first interesting idiot-proof tile (IPT) is the L-tromino.

A question which I like to ask sixth-formers at their interview for University is to evaluate

not worrying for the moment about the behaviour at x = 0 or x = 1, or the limits of the integral. This question is difficult enough for some students but on a good day one can lead them through to the correct solution (log|log x|) by various suggestions like “try a substitution” (x = et) or “do you know a rule for when the top is the derivative of the bottom?”