4. A combinatorial zoo

‘A combinatorial zoo’ presents a menagerie of combinatorial topics, ranging from the box (or pigeonhole) principle, the inclusion–exclusion principle, the derangement problem, and the Tower of Hanoi problem that uses combinatorics to determine how soon the world will end to Fibonacci numbers, the marriage theorem, generators and enumerators, and counting chessboards, which involves symmetry. The method used to average the numbers of colourings that remain unchanged by each symmetry in this latter problem is often called ‘Burnside’s lemma’. This concept has since been developed into a much more powerful result, which has been used to count a wide range of objects with a degree of symmetry, such as graphs and chemical molecules.

8. Designs and geometry

Block designs are used when designing experiments in which varieties of a commodity are compared. ‘Designs and geometry’ introduces various types of block design, and then relates them to finite projective planes and orthogonal latin squares. A block design consists of a set of v varieties arranged into b blocks. If each block contains the same number k of varieties, each variety appears in the same number r of blocks, then for every block design we have v × r = b × k. A balanced incomplete-block design is when all pairs of varieties in a design are compared the same number of times. A triple system is when each block has three varieties.

7. Square arrays

‘Square arrays’ is concerned with magic squares and latin squares. An n × n magic square, or a magic square of order n, is a square array of numbers (usually the numbers from 1 to n 2) arranged so that the sum of the numbers in each of the n rows, each of the n columns, or each of the two main diagonals is the same. A latin square of order n, is a square array with n symbols arranged so that each symbol appears just once in each row and each column. Orthogonal latin squares are also discussed along with Euler’s 36 officers problem.

6. Graphs

Graph theory is about collections of points that are joined in pairs, such as a road map with towns connected by roads or a molecule with atoms joined by chemical bonds. ‘Graphs’ revisits the Königsberg bridges problem, the knight’s tour problem, the Gas–Water–Electricity problem, the map-colour problem, the minimum connector problem, and the travelling salesman problem and explains how they can all be considered as problems in graph theory. It begins with an explanation of a graph and describes the complete graph, the complete bipartite graph, and the cycle graph, which are all simple graphs. It goes on to describe trees in graph theory, Eulerian and Hamiltonian graphs, and planar graphs.

2. Four types of problem

‘Four types of problem’ explains that combinatorics is concerned with four types of problem: existence problems (does x exist?); construction problems (if x exists, how can we construct it?); enumeration problems (how many x are there?); and optimization problems (which x is best?). Existence problems discussed include tilings, placing dominoes on a chess board, the knight’s tour problem, the Königsberg bridges problem, the Gas–Water–Electricity problem, and the map-colour problem. Construction problems include solving mazes, and the two types of enumeration problems considered are counting problems and listing problems. Examples of an optimization problem include the minimum connector problem and the travelling salesman problem. The efficiency of algorithms is also explained.

5. Tilings and polyhedra

A tiling of the plane (or tessellation) is a covering of the whole plane with tiles so that no tiles overlap and there are no gaps. Polyhedra are three-dimensional solids that are bounded by plane faces. How many are there, and can we construct and classify them? ‘Tilings and polyhedra’ describes the different types of tilings and polyhedra that are possible, beginning with regular tilings made up of regular polygons, semi-regular tilings, and irregular tilings. There are only five types of regular polyhedra—the tetrahedron, cube (or hexahedron), octahedron, dodecahedron, and icosahedron—but there are numerous semi-regular polyhedra, including prisms and antiprisms.

3. Permutations and combinations

Permutations and combinations have been studied for thousands of years. ‘Permutations and combinations’ considers selecting objects from a collection, either in a particular order (such as when ranking breakfast cereals) or without concern for order (such as when dealing out a bridge hand). It describes and investigates four types of selection—ordered selections with repetition, ordered selections without repetition, unordered selections without repetition, and unordered selections with repetition—and shows how they are related to permutations, combinations, the three combination rules, factorials, Pascal’s triangle, the binomial theorem, binomial coefficients, and distributions.

1. What is combinatorics?

Combinatorics can loosely be described as the branch of mathematics concerned with selecting, arranging, constructing, classifying, and counting or listing things. ‘What is combinatorics?’ explains that the subject can be dated back some 3000 years to ancient China and India. For many years, especially throughout the Middle Ages and the Renaissance, it consisted mainly of problems involving the permutations and combinations of certain objects, but over the succeeding centuries the range of combinatorial activity has broadened greatly. Combinatorics now includes a wide range of topics such as the geometry of tilings and polyhedra, the theory of graphs, magic squares and latin squares, block designs and finite projective planes, and partitions of numbers.

9. Partitions

How many ways can a number be split into two, three, or more pieces? ‘Partitions’ considers this interesting problem and the way in which Leonard Euler started to investigate them around 1740. Euler considered the generating function of the sequence of partition numbers and devised his pentagonal number formula. His publication Introduction to the Analysis of Infinities in 1748 outlined the difference between distinct and odd partitions. Many mathematicians worked on the partition problem, but it was not resolved until G. H. Hardy and his collaborator Srinivasa Ramanujan in 1918 published an exact formula for partition numbers using a new method in the theory of numbers called the ‘circle method’.