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.

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.

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.

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.

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.

How do you know that the equation x3 + x = 5 has one real root? Well, the function x3 + x is monotonic increasing: any increase in the value of x produces an increase in x3 + x. In particular, since 13 + 1 = 2 and 23 + 2 = 10 , we know that, for some x between 1 and 2, x3 + x = 5 . The function must “pass through” 5, and, since it is always increasing, it does this exactly once.

The distribution of the prime numbers is of course a well-trodden path. The casual enquirer, working with small numbers, soon finds that the distribution has no obvious regularities, and sees that the primes thin out. The Prime number theorem (which is not elementary) helps to firm up this observation:

Prime number theorem: If π(n) is the number of primes no greater than n, then i.e. π(n)/(n/ln(n)) → 1 as n → ∞. Thus for the proportion of prime numbers upto to n, π(n)/n, we have that which tends slowly to 0.