The picture of a rainbow on the front cover of this issue of the Gazette was taken by Nicholas Robins in May 1993. He was mystified that the rainbow was upside down, and wrote to the ‘Notes and Queries’ column in the Guardian asking if anyone could explain why. From subsequent correspondence and discussion it emerged that there were further irregularities. Firstly, it was not raining. Rainbows are caused by refraction of light through spherical raindrops, so how could you have a rainbow with no raindrops? Secondly, this rainbow was in the same direction as the sun. To see a normal rainbow you have to stand with your back to the sun.

Tetris is a computer game which has obsessed many computer users and attracted much attention, despite the simplicity of its rules. This paper addresses the question: ‘can you “win” the game Tetris?’

Designed by Soviet mathematician Alexey Pazhitnov in the late eighties and imported to the United States by Spectrum Holobyte, Tetris won a record number of software awards in 1989. Versions of Tetris are sold for most personal computers. There are Tetris arcade games, Tetris Nintendo cartridges, and hand-held Tetris games; Tetris has been played on machines ranging from mainframes to calculators. The game's success has prompted the invention of several similar games, including Hextris, Welltris, and Wordtris.

We propose an interpretation of the integers as operations on permutations. Addition and multiplication of integers represent operations on these operations. It follows that an equation involving integers, addition and multiplication is valid if and only if the corresponding operations on permutations are the same. In our interpretation, the validity of the fundamental laws of calculation is an immediate consequence of well-known properties of permutations. For example, the ‘Law of signs’: (–1) x (–1) = 1, is valid because the inverse of the inverse of any permutation is the permutation itself.

The succession of enlargements from ℤ (the system of the integers) to ℚ (that of the rational numbers) and thence to ℝ (the real numbers) and then ℂ (the complex numbers) is generally seen as one of increasing richness of structure. But gains are often accompanied by losses: for instance the step from ℝ to ℂ leads to the glories of complex analysis but away from ordinal properties once thought essential to any concept of number.

As far as group structure is concerned there are gains and losses at each stage of the progression. The following exposition is intended to demonstrate this without assuming more than a minimal knowledge of groups and of the theoretical foundations of analysis. Beyond some elementary arithmetical algebra, and an acquaintance with standard notation and vocabulary for maps in general, essentials taken for granted include the idea of a homomorphism (of groups), the existence and fundamental properties of the exponential function (of a real variable), and the Fundamental Theorem of Algebra (which is actually a theorem in complex analysis).

I have always disagreed with government ministers who assert that it is a good idea to separate teaching from research at the universities. For I believe that with mathematics, research and good teaching go hand in hand. And if we interpret ‘research’ in its widest terms of questioning, discovery, involvement, etc, then I believe that the same holds true at the school level. Mathematics needs to be taught as a vibrant, living subject which challenges the intellect and the imagination, not as a dusty collection of historical facts. Further, asking questions is just as much fun as answering them!

Planarity is a concept that appears in most discrete mathematics courses (in particular, it is a syllabus item in all A level Discrete mathematics syllabuses). I have recently taught a unit in graph theory to a group of BEd students and, when I came to the concept of planarity, I decide to approach it in a different way – through the context of Cabri Géomètre, a dynamic geometry package. In the article I would like to share some of my ideas and reflections on this approach to teaching this topic, paying particular attention to the role of Cabri Géomètre in scaffolding the important ideas involved in an understanding of planarity.

Integer Geometry is the geometry of points that are pairwise integer distances apart. An integer polygon is a convex set of n points in the plane such that no three points are collinear and the distance between any two points is an integer. Similarly an integer polyhedron is a convex set of n points in space such that no three are collinear, each face is an integer polygon, no two faces are coplaner, and the points are all integer distances apart. The measure used to order such polygons and polyhedra is perimeter-plus which is the sum of all of the edges added to the sum of all of the diagonals [1]. The purpose of this paper is to provide pictures of some fundamental examples as well as a brief explanation of some of the constructions. We begin by looking at polygons and then use those to build polyhedra and conclude by examining two closely related examples of sixfaced polyhedra.

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.

When a parent sees a little girl sitting on the floor cutting paper dolls, many thoughts may come to mind: ‘She’s keeping out of trouble’ or ‘She’s making a mess’ or even ‘There go my tax returns’. The thought that should have come to my parent’s mind, however, was ‘One day she’ll be a mathematician’. My grandmother, who worked as a dressmaker, often allowed my sister and me to use her razor sharp scissors on the strips of leftover tracing paper. This paper is inspired by a notebook that I kept in grade school when I ‘studied’ paper dolls, and the figures are based on dolls found pressed between the pages.

It is common today for students in elementary calculus to possess a powerful graphics calculator, such as the TI-82. These machines have had a dramatic impact on how teachers present the subject. The teacher population is clearly divided when it comes to the emphasis on the use of the calculators. There are instances, though, where it is difficult to argue with the calculator's effectiveness as a teaching tool in calculus. One such case is in using Newton's method to find the roots of an equation, for once the algorithm is presented, it is merely a matter of performing the numerical calculations for the sequence of iterates, and this is an operation that the TI-82 does very well.