Essays on recreational mathematics

'Euclidean geometry' sounds like a heavy term, but it simply describes what we think of as 'normal' geometry - geometry that holds true in the familiar two-dimensional (or Euclidean) plane. In this geometry, many concepts which we think of as 'common sense' hold true. But non-Euclidean geometries exist too. And they are very important!

Sudoku is a logic-based number-placing puzzle. The goal is to fill a 9 x 9 grid with digits so that each column, each row, and each of the nine 3 x 3 sub-grids contain all the digits from 1 to 9, each precisely once in every sub-grid. How many variations of the 9 x 9 grid exist? And how are puzzles formulated and graded?

Place a number of objects anywhere in a plane. For each object there is a region consisting of all points of the plane closer to that object than to any other. The regions associated with each object with this proximity property are called Voronoi cells. They turn out to have many, and often surprising, real world applications.

Within the field of campanology, 'change ringing' is the art of ringing a set of tuned bells in a carefully controlled manner to produce precise variations in their successive striking sequences, known as 'changes'. The task demands dexterity and concentration, and is endlessly fascinating for mathematicians too.

Within the infinity of prime numbers lie the noble class of Mersenne primes. They are rare, they have a particular beauty, an elegant simplicity, and they describe the very largest of all known prime numbers. They have been, and continue to be, centre stage in a long-standing search to find larger and larger primes.

Fractals are objects, including geometrical shapes, which exhibit similar patterns at progressively smaller scales. Approximations in nature include snow flakes, lightning flashes, and river networks. In mathematics, the most well-known, and arguably most spectacular, is the fascinating 'Mandelbrot set', and the closely related 'Julia sets'.

Leonhard Euler, 1707-1783, was a Swiss mathematician and physicist, who made important discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to branches such as number theory and topology. He is held to be one of the greatest mathematicians in history.

The number e is a mathematical constant approximately equal to 2.718. It is one of the most important in mathematics. It was first described by Swiss mathematician Jacob Bernoulli, in 1683, in his studies of compound interest, and later by Leonhard Euler. It turns out to be ubiquitous across mathematics and the natural sciences.

Everyone is familiar with 'modulo arithmetic', even if they don't know the term. It is the basis of our measurement of time: we measure hours to modulo 12 (or 24), days of the week to modulo 7, the months of the year to modulo 12, and the years of the century to modulo 100. There are some entertaining consequences for calendars.

This collection is not an examination of each and every number, but I have picked out a few to illustrate the kind of properties that number theorists and recreational mathematicians enjoy examining. Here we will consider some of the properties of the number 23, and a few of the places where it crops up.

Pierre de Fermat asserted, but could not prove, that every number of the form F(n)=2^(2^n)+1 is prime. The first five values are indeed prime. But they quickly become much harder to verify, because they increase in size so rapidly. It turns out that there are no known prime Fermat numbers beyond F(4). Others might exist. They might not.

A Platonic solid is a regular convex geometrical body. Each of its faces are identical regular polygons, with all angles and all sides being equal, and with the same number of faces meeting at each vertex. Just five solids meet these criteria. Allowing more than one type of regular polygon to meet at each vertex leads to the 13 Archimedean solids.

The five Platonic solids satisfy the conditions of complete regularity in terms of faces and sides. Allowing more than one type of regular polygon to meet at the vertices leads to a further 13 Archimedean solids. Relaxing the condition that the polyhedron must be entirely convex, leads to the Kepler-Poinsot polyhedra.