Some public libraries use a barcode on the date slips in their books which is easily cracked by anyone with a knowledge of binary arithmetic, but those with which we are now so familiar on purchased goods are much more of a puzzle, as there appears to be little correlation between the printed digits and the bars by which they are represented – a little trial will reveal that it is certainly not one-one!

Even with a large number of examples to experiment with, I very much doubt if anyone not in the secret would succeed in solving the problem, which it is the object of this article to elucidate.

Everyone has seen barcodes, on groceries, newspapers and virtually every other product that they buy, but how many people know what they mean? I have always felt that one day I ought to sit down and work out what the various stripes in the barcodes meant – they were an obvious challenge. However, on the various occasions when I have tried, I have had no success, and I have given up (and as you will see from what follows, the system is sufficiently complex that a simple attack on the coding system is unlikely to succeed).

Twenty years ago one of the plots in the TV series Mission Impossible involved a chess tournament in which Leonard Nimoy (without the pointy ears, but in every other respect identical to his Star Trek persona) won a game by obeying the instructions of a computer relayed over an earphone. While many of the antics of the Impossibles remain … impossible, it is only be a matter of time until chess computers can defeat every human being on the planet. However a chess positioninvented by David Norwood [Daily Telegraph, Weekend p.28, 7 Aug 1993] shows that the game may not yet be over for homo sapiens. The position in figure 1 could occur in actual play, but is most unlikely to.

The crisis in applicable maths for schools

In the first paper in this series I remarked that there have been three successive paradigms which have dominated school mathematics since 1960: three “eras” of predominant pedagogy –“traditional” maths, “new” maths and “applicable” maths. The main theme of the first paper was, however, that the new degree of involvement of school mathematics with applications – which came about as a result of the third paradigm – can hardly be unlearned. It will be very difficult, if not impossible, to disengage from paradigm 3. Society is most unlikely ever to let school mathematics revert to a condition of explicit, high-academic, mandarinstyle leadership, whether of the traditional pure mathematics or the modem pure maths allegiance. Yes, the third paradigm looks as if it may be beginning to falter, but the reason, I suggest, is that we are not getting the full pedagogic benefit from the particular genre of applications we are using, and from the particular style of application-mindedness we are building into our teaching.

Classically, the force of gravity between two masses m and M is given by the Principle of Universal Gravitation or the inverse-square law

where x is the distance between the centres of mass, and G is the universal gravitation constant (approximately 6.67x10–11 m3 kg–1s–2). In elementary calculus, a common application is to assume that the acceleration due to gravity near the Earth’s surface is a constant, and then to do simple velocity and position calculations. If instead we apply the above law of force, then the acceleration due to gravity of an object of mass m is a function of the distance x = x(t) beween its centre of mass and the centre of the Earth at time t. Then, on applying F = ma

so the velocity and position calculations will be different. The purpose of this note is to present these calculations and their interesting consequences in a manner appropriate for a first year calculus course. (See also Temple B., and Tracey, C.A., “From Newton to Einstein”, American Mathematical Monthly, 99, 507–521,1992).

The following is a well known pursuit problem: n bugs are at rest at the vertices of a regular polygon. At a certain instant each bug starts crawling towards its neighbour on the right with a constant speed, always altering its course so as to be headed directly towards its neighbour. How long does it take for the bugs to meet at the center of the polygon and what paths do they trace out in getting there? The earliest reference I could find to this problem was in a collection of mathematical puzzles edited by L.A.Graham. A more recent reference to it occurs in the amusing and informative collection of geometrical curiosities by David Wells. The “n-bug” problem was also discussed briefly in two of Martin Gardner’s columns in Scientific American. The cover of the July 65 issue of that magazine shows the equiangular spirals traced out by four bugs as they “square dance” their way towards their tryst at the centre.

This article provides a brief introduction to Evangelista Torricelli and to one aspect his work. It is divided into three parts. Part 1 gives some biographical details and a short survey of his work and achievements. Part 2 is concerned with a geometrical approach towards the study of projectiles as seen in Torricelli's early work De motu. Part 3 shows how Torricelli applied his projectile studies to practical ballistic problems.