Over the years we have come across many different methods of proving the sine function addition formula

and we have long felt it could be both interesting and useful to gather together in one article a list of methods to illustrate how different ideas from geometry, algebra and calculus may be used to prove it.

In a recent issue of the Gazette S. Parameśwaran defined a positive integer n to be an S•P number if it is the product of the sum of its digits and the product of its digits; for example, 135 and 144 are S•P numbers because

His conjecture that there are only finitely many such numbers was proved in where it was shown that any S•P number is less than 1060. Here, we prove the following result.

The Lyness equation (1)

\begin{equation}{X_{n + 1}} = \frac{{{X_n} + a}}{{{X_{n - 1}}}},\,(a,{x_1},{x_2} > 0)\end{equation}

The history of cryptography is punctuated by the invention of clever systems to encipher messages and, sometime later, equally clever systems for cryptanalysing the enciphered messages to determine their meaning. Most enciphering schemes of any worth enjoy a relatively lengthy period of prominence before sufficiently determined cryptanalysts undermine their security by figuring out how to attack them. In response, cryptographers devise new and improved schemes and then the cycle repeats. Cryptographers have learned from history that it is dangerous to declare any enciphering scheme unbreakable; at best they are considered to be very secure. But there is one scheme, a scheme that has been around since 1917, that truly is unbreakable. It is the perfect cipher.

The first part of this article covers some well-known ground in preparation for the second part on the Archimedean honeycomb duals. Mathematically speaking some of this second part is, so far as I know, no more than conjecture: I do not myself know of a proper systematic approach to such honeycombs.

As far as possible I have used names for the polyhedra from standard references (e.g. [1]). But references vary, and I have not found all of them described. In such cases I have tried to choose names which are appropriate and technically correct.

Everyone connected with the teaching or practice of mathematics is aware that in recent years there has been a great increase in the number and variety of opportunities for the employment of mathematicians. In particular, industry, and especially engineering industry, has found a need to recruit mathematicians in large numbers with a view to employing them to do mathematics as a full-time occupation. This is quite a new development; although the value of the mathematically trained mind has long been recognised in many places, it is only quite recently that mathematics, as opposed to mathematicians, has been extensively employed by industry This change has, of course, largely coincided with the advent of the electronic digital computer, and the computer has certainly had a great influence on it; new mathematical disciplines have grown up as a result of the existence of computers, and it is now practicable to solve problems whose solution would be inconceivable without the use of a computer. As a result of this it has become fashionable to attribute the spread of mathematics to the computer. I think it more likely that the computer has accelerated rather than created the demand for mathematicians, and that the development of new industries and the introduction of new techniques to old ones would have brought about a demand for mathematicians even in the absence of computers. However there is no doubt that this demand exists, and it has certainly been stimulated by the existence of the computer.