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.

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.

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.

This extremum problem is really a classical beauty. It has had a long and interesting history since it was formulated by Fermat in the 17th century. Given three points A, B and C, the task is to find a point P such that the sum of distances PA + PB + PC is minimal; (see Figure 1). After a few years Torricelli found the solution: P should be situated so that the angles between the half-lines PA, PB and PC are all 120° (except when one angle of the triangle ABC is greater than or equal to 120°). The solution has then been rediscovered many times in new and interesting ways.

How many ways are there of making up £1 using coins? Omitting the use of the £1 coin, we are asking for the number of ways of ‘partitioning’ 100 using parts of 1, 2, 5, 10, 20, 50. For n ≥ 0, let f (n) denote the number of solutions to the equation

in non-negative integers xi, so that our required number is f (100). It is not difficult to see that f(n) is the coefficient of tn in the product Π(1 + tk + t2k +… ) over the parts k = 1, 2, 5, 10, 20, 50.

There are hints of unique factorisation in Greek arithmetic. Indeed, some commentators have seen the Fundamental Theorem of Arithmetic (FTA), that the natural numbers can be expressed as products of primes in a unique way, lurking in Euclid’s Elements (c. 300BC). What can be said with certainty is that the history of the FTA is strangely obscure. It is not too much of an exaggeration to say that the result passed from being unknown to being obvious without a proof passing through the head of any mathematician.

In the High and Far-Off Times, O Best Beloved, before GCSE, frogs did not spend their days in queue-jumping investigations. Each frog sat on a lily-pad in the middle of his very own pond, wondering how to get to the edge.

Now these frogs were Progressive Frogs—more than that, O Best Beloved, they were Geometrically Progressive Frogs, and each time they jumped, they could only manage to jump half the distance of their last jump.

In introductory courses in abstract algebra the cyclic groups Zn, the dihedral groups Dn, and the symmetric and alternating groups Sn and An, for various integers n, are often used to provide examples of various concepts. The intent of this note is to argue that another group could readily be added to this list: SL (2, 3), the multiplicative group of two by two invertible matrices of determinant 1 with entries from the field with three elements. Though this non-abelian group of order 24 does not appear often in introductory presentations, it is an excellent source of group theoretic examples and is also accessible to beginners, who at this stage of their study most likely have had some experience with matrices. Thus computations in SL (2, 3) can be implemented almost as easily as in the cyclic groups, and defining relations need not be introduced.

Much of the investigational work that pupils do in schools reduces to a routine of generating data, spotting number patterns and describing those patterns [1]. I will refer to this routine as the inductive approach. Many GCSE candidates only use algebraic notation as an alternative translation of what they have already articulated in words. Yet surely the power of algebra is in its potential for providing proof and suggesting further enquiry, based on the similarity of structure between the algebraic model and the original situation.