Tony Gardiner

An introduction to combinatorics

Introduction Permutations and Combinations The Inclusion-Exclusion Principle Partitions Stirlings Approximation Partitions and Generating Functions Generating Functions and Recurrence Relations Permutations and Groups Group Actions Graphs Counting Patterns Polyas Theorem Solutions to the Exercises Suggestions for Further Reading List of Symbols Index

Triangular numbers and perfect squares

The n th triangular number is defined to be the sum of the first n positive integers: Thus In a letter to Mersenne in 1638 [1, p. 61], Fermat claimed that every positive integer can be written as a sum of at most three triangular numbers. This remarkable result was eventually proved by Gauss in 1796, at the age of 19.

Learning to prove: using structured templates for multi-step calculations as an introduction to local deduction

It is generally accepted that proof is central to mathematics. There is less agreement about how proof should be introduced at school level. We propose an approach—based on the systematic exploitation of structured calculation—which builds the notion of objective mathematical proof into the curriculum for all pupils from the earliest years. To underline the urgent need for such a change we analyse the current situation in England—including explicit evidence of the extent to which current instruction fails even the best students.

Triangles and Tetrahedra, Quadrilaterals and Cubes: A Classroom Investigation

One way to make a tetrahedron is to use four equilateral triangles. Let’s see why this works. When we fit two equilateral triangles of the same size together along an edge, the edges match up.

An Outline of Set Theory

One Projects.- 1. Logic and Set Theory.- 2. The Natural Numbers.- 3. The Integers.- 4. The Rationals.- 5. The Real Numbers.- 6. The Ordinals.- 7. The Cardinals.- 8. The Universe.- 9. Choice and Infinitesimals.- 10. Goodsteins Theorem.- Two Suggestions.- 1. Logic and Set Theory.- 2. The Natural Numbers.- 3. The Integers.- 4. The Rationals.- 5. The Real Numbers.- 6. The Ordinals.- 7. The Cardinals.- 8. The Universe.- 9. Choice and Infinitesimals.- 10. Goodsteins Theorem.- Three Solutions.- 1. Logic and Set Theory.- 2. The Natural Numbers.- 3. The Integers.- 4. The Rationals.- 5. The Real Numbers.- 6. The Ordinals.- 7. The Cardinals.- 8. The Universe.- 9. Choice and Infinitesimals.- 10. Goodsteins Theorem.

Mathematical Method: Does It Exist?

Serious science students and most science teachers are more or less aware of something called ‘scientific method’, and the fact that it depends on a subtle interplay involving mental constructs in the form of ‘theory’, on the basis of which one can make novel ‘predictions’ (as opposed to retrospective ‘explanations’), which can then be corroborated, or refuted, by ‘experiment’. The mental image which people have of this scientific method is often garbled, but the scientific trinity of theory, prediction, and experiment has become a commonplace. (See, for example, [17], especially pages 42–46.)