A recent excellent book by Paul J. Nahin on Oliver Heaviside, a self-taught genius of the Victorian age, contains several nice examples of problem solving by the master. Two of these, concerning the flow of electricity along telegraph cables, are suitable for sixth formers and are described here. In passing it may also be mentioned that these cables, used first for telegraphic signals and later for voice signals, were a triumph of both the practical and theoretical skills of the nineteenth century, and were in use for world-wide communications for over a hundred years up to the nineteen seventies.

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.

In an article in a recent edition of Mathematics in School (May 1989), Ann Kitchen considers the mathematics of lifting bridges, among which is the bascule bridge shown below, where the counterpoise weight travels down the arc of a circle. This was chosen as a “simplification”, though even so an explicit analytical solution cannot be found. In her example the potential energy of the system was chosen to be the same in the closed and open positions, and there is then a position of unstable equilibrium in between.

One way to make a tetrahedron is to use four equilateral triangles. Let’s see why this works. When we fit two equilateral triangles of the same size together along an edge, the edges match up.

My younger son has become interested in the game “dungeons and dragons”. He is intrigued by the shapes of the dice that are used to play the game and by the number of faces they have. Besides the standard cubical die with its 6 faces, there are the other regular solids with 4,8,12 and 20 faces. There is also a die with 10 faces, each in the form of a kite, and my informant points out longingly that our local games shop has a die with 100 faces. “Could you have one with an odd number of faces?” he asks.

His innocent question led me to think about many mathematical aspects of dice. This article answers his question, examines the geometry of dice (highlighting some unnamed solids which deserve to be better known to model-makers), considers surprising aspects of the mechanics of tossing dice and describes sets of unusual dice whose probabilities offer scope for many games which can trap the unwary gambler.

A standard introductory calculus problem is to calculate the size of small squares that have to be removed from (say) a unit square of metal so that the resulting net folds into a tray of maximum volume. With notation as in the figure, this requires that be maximised which occurs when x = 2/3 giving a maximum volume of 2/27. (Surprisingly, this conclusion may also be reached without using calculus via the inequality of arithmetic and geometric means:

with equality if and only if or x = 2/3.)

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.

The article describes an accessible example of dynamical chaos which has been discussed recently in the Journal of Physics. It is simple enough to be explored by microcomputer at home or school. It concerns a point “billiard ball” bouncing round an ideal square “billiard table” that rotates steadily. Without rotation the ball's motion would be a perfectly regular sequence of straight-line transits; with rotation most of the motion is unpredictably jumbled—chaotic, as we'll see. First some simple theory is needed.

A partition of n is a decomposition of n into a sum of positive integers. So the 5 partitions of 4 are

The number of partitions of n is written p(n) and (with the convection that p(0) = 1) the sequence p(0), p(1), p(2), ... begins

1, 1, 2, 3, 5, 7, 11, 15,....

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?”

Over the years there have been various articles in the Gazette [1, 2, 3] concerning the running time of cassette tapes, but none of them uses the property described here, although it is relevant to all of them.

Let us assume that the revolution counter on a cassette or video recorder is operated by the run off spool. Let a be the radius of the full spool and r the radius when an angle θ radians has been unwound. Rather as in [2] it is clear that a reasonable continuous model of this situation satisfies the equation

When students work through a written proof what is it that they are attending to? At what level are they taking in and making sense of what they see? The mathematical symbols (i.e. diagrams, equations, etc) that are found in proofs are a result of the writer seeing some generality, attempting to express that generality and to convince themselves and others of its validity. We recognise that students, when reading a proof may work directly with the symbols, somebody else's symbols, without attempting to enter into the process of expressing generality for themselves.

With the arrival of calculators having graphical displays, students at Alevel have been able to investigate the graphs of various functions where previously the analysis involved would have prevented them from doing so. Unfortunately (or fortunately?) the technology on most hand-held calculators and even on many graph-plotting packages for standard micros is not capable of handling all functions which A-level candidates might like to investigate.

We have both used the iteration x0 = 1, xn+1 = cxn (n ≥ 0) with groups of students to investigate the range of values of c > 0 for which the iterative schema converges. It is well known that there are real numbers p, q with 0 < p < q which divide the domain of positive reals into three intervals in which three distinct kinds of behaviour are exhibited. For values of c in T = (0, p) the iteration settles down to oscillations between two real values of x. For values of c in S = (p, q) the iteration converges to a single real value of x, and for values of c in D = (q, ∞) the iteration diverges. Approximations to p and q can be found by numerical investigations with calculators and computers, and the derivation of analytic expressions for p and q is within reach of many sixth-formers.