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.

Among the very few reported observations of singing angels, the account given by the shepherds of the angelic choir on Christmas night records some of the words but gives no account of the music, even though shepherds are much celebrated as pipe players and so form an unusually musical profession. But in which key did the angels sing? This question could only have been answered for us by a shepherd with perfect pitch, that is, an unusually good ear for pitch (how high or low the music is). However, the question also contains the hidden assumption that the angels based their music on the key system, which was not known on earth until the sixteenth century AD, so the question should be changed to “Which musical system did the angels use?”. Clearly, as angels are perfect, the system must be the best available, so we may be able to answer the question theoretically.

“A derivative is a linear transformation which approximates a function near to a point.” I heard this definition for the first time in 1964, in a lecture at Oxford on the calculus of several variables (the lecturer was D.A. Edwards). The words came as a revelation, and from then on I had a new conception of differential calculus. At school, I thought that I knew about calculus because I could write out proofs of differentiation ‘from first principles’. I enjoyed going through the hoops and proving that δy/δ tends to a limit, but somehow the idea of differentiation never became ‘part of me’.

I believe many people have been inspired by Grunbaum and Shephard's book on tilings and patterns. On reading the book, as a computer programmer, I thought it would be a challenge to display the tiling patterns in colour on the screen of my home computer. Not having a hard disc, I had found the storage for fractals too demanding, notwithstanding the time needed to compute the intricate patterns. Hence the idea of producing a variety of interesting patterns without such overhead was attractive.

The editor of this journal recently deplored the inability of members of the government to make correct use of the simple mathematics that they encounter in their various capacities. The purpose of this article is to call attention to a common mathematical failing of those who are “authorities” in questions such as the life expectancies of non-renewable resources. My particular example comes from Britain, although failures such as this one are common among “experts” in the U.S.

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).

An intriguing family of tile patterns can be generated by applying the self-replicating geometry of a pair of isosceles triangles called the golden triangles. The patterns have an aesthetic appeal and have geometric properties which might be investigated further. The family includes patterns which were discovered by Roger Penrose in the 1970s, and which received much attention when they were proposed as a model for quasicrystals. The use of golden triangles to obtain tile patterns stems from the work of R.M. Robinson cited in [2, p540] and extends the investigations of [3]. The method employed in obtaining the patterns is experimental mathematics; geometrically defined experiments are performed using a computer and the results examined for their properties.

An {r, s}-leaper is a generalized knight that can jump from (x, y) to (x ± r, y ± s) or (x ± s, y ± r) on a rectangular grid. The graph of an {r, s}-leaper on an m x n board is the set of mn vertices (x, y) for 0 ≤ x < m and 0 ≤ y < n, with an edge between vertices that are one {r, s} -leaper move apart. We call x the rank and y the file of board position (x,y). George P. Jelliss raised several interesting questions about these graphs, and established some of their fundamental properties. The purpose of this paper is to characterize when the graphs are connected, for arbitrary r and s, and to determine the smallest boards with Hamiltonian circuits when s = r + 1. or r = 1. (A Hamiltonian circuit is a closed path that visits every cell exactly once.)

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.

We are so used to using calendars that we do not often ask when they were first used and why they were used. One of the earliest places where calendars are known to have been in use is ancient Egypt. Egyptian agriculture was based on using the Nile flood water. The Nile floods at the same time every year but the cause of the flood is not connected with the weather in Egypt itself. It was therefore essential to be able to predict the time of the Nile flood in advance if advantage is to be made of the flood waters. However this is a special situation. For example when deciding when to prune my roses in the spring I could consult a calendar. However if I had no calendar I could easily find the right time of year for pruning the roses by observing when some other plant had reached a certain stage of development. In fact given the difference between one season and another this method is probably better than using a calendar anyway.

A correspondent raised a question concerning Premium Bonds. (See page 459 of the November 1992 Gazette.) He had held 10,000 bonds for 2 years without winning a single prize and wondered how likely this was. In this note, I will describe a simple method for calculating the probabilities of winning. I will also look at the effects of some recent changes in the Premium Bond rules.

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.

I was intrigued by Brian Bolt’s note “The Home Stretch”, as it had not occurred to me to look at Steiner trees spanning networks in three dimensions before. Nor did I find any references to them in the (admittedly limited) literature I consulted, though plenty has been written about the two dimensional case (see, eg, Gardner, Wells, MacKinnon). I was quickly able to improve on Bolt's tree, and decided to investigate networks for other shapes.

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).

Here is a published example of what I mean by the sinusoidal fallacy: