The *Plus* teacher packages are designed to give teachers (and students) easy access to *Plus* content on a particular subject area. Most *Plus* articles go far beyond the explicit maths taught at school, while still being accessible to someone doing A level maths. They put classroom maths in context by explaining the bigger picture — they explore applications in the real world,
find maths in unusual places, and delve into mathematical history and philosophy. We therefore hope that our teacher packages provide an ideal resource for students working on projects and teachers wanting to offer their students a deeper insight into the world of maths.

## Proofs

The notion of proof lies at the very heart of maths — without it, maths would be little more than a vaguely interesting collection of computational tools. It's when it comes to proving things that mathematicians let lose their genius and creativity, and in the process often discover unexpected surprises or deep philosophical issues. But proofs can also be daunting. So to help you and your
students along, we've brought together a range of *Plus* articles on proofs, grouped together in the following categories:

- Proofs: what are they and why do we need them?: These articles explore the notion and role of proofs through the ages, with many interesting examples.
- Nice proofs: The articles in this category contain explicit examples of nice, fun, or surprising proofs, put into historical, cultural, and mathematical context.
- Elusive proofs: In this category we look at proofs that have eluded mathematicians for centuries, exploring some famous unsolved (or recently solved) problems, and ways of attacking them.
- Philosophical proofs: It may seem as if maths is all about certainty, but there are actually many philosophical questions surrounding what constitutes a proof. What's more, there's proof that not everything can be proved. The articles in this category explore these limitations and questions.

Don't forget that our sister site NRICH has many hands-on problems, activities and articles on proofs.

## Proofs: what are they and why do we need them?

Why we want proof — What are mathematical proofs, why do we need them and what can they say about sheep?

The origins of proof — Part I — This article explores deductive reasoning and looks at the earliest known example of a proof.

The origins of proof — Part II — This article explains how the notion of proof was brought from mathematics into physics by the mathematician and astronomer Johannes Kepler.

The origins of proof — Part III — For millennia, puzzles and paradoxes have forced mathematicians to continually rethink their ideas of what proofs actually are. This article explains the tricks involved and how great thinkers like Pythagoras, Newton and Gödel tackled the problems.

The origins of proof — Part IV — This article explores what a proof really *is*, and how we know that we've actually found one. One for the philosophers to ponder...

1089 and all that — Mathematics is full of surprises, which often only reveal themselves when you're trying to prove something. This article explores some examples.

## Nice proofs

Seeing Pythagoras — The great thing about geometry is that you can sometimes prove things using pictures. Enjoy these three great visual proofs of Pythagoras' theorem!

Some lovely proofs by picture — Don't like trigonometry? Don't worry, here are three beautiful proofs of a well-known result that make do without it.

Intriguing integrals part I and part II — These two articles revisit an old favourite, the integral of x^{k}.

Dividing Walls: Topology and topography part I and part II — These two articles explore how topology can overcome topography, giving some nice proofs in the process.

On the dissecting table — This article looks in detail at a geometric proof of Pythagoras' theorem.

Euler's polyhedron formula — One of Leonhard Euler's many contributions to maths is a surprising and simple geometric formula relating the number of edges, vertices and faces of a polyhedron. This article explores the formula and its proof.

Friends and strangers — Sometimes order can be a consequence of disorder. This article looks at the colourful world of Ramsey theory, having a go at proving some bewildering facts.

Looking out for number one — You might think that if you collected together a list of naturally-occurring numbers, then as many of them would start with a 1 as with any other digit, but you'd be quite wrong. This article looks at the so-called Benford's law and proves it too.

Sundaram's sieve — The prime numbers are the atoms amongst the integers, and while we know that there are infinitely many of them, there's no general formula that generates them all. This article looks at an astonishingly simple and little-known algorithm that sieves out all primes up to a given number, and guides you through a proof that it works.

An infinite series of surprises — Infinite series provide some of the most counter-intuitive surprises in mathematics. This article shows how to go about proving the convergence (or divergence) of some of these beasts.

John Conway – discovering free will — In 2004 John Conway and Simon Kochen proved a very controversial theorem called *the free will theorem*. This three-part article explores the theorem and its proof.

Maths aMazes — This article uses a proof to find a way out of a maze. And it leads us into the dark territory of murder, suicide, adultery, passion, intrigue, religion and conquest...

Mathematical mysteries: painting the plane — Your task is to paint the plane so that every point is coloured, but so that any two points on the plane which are exactly 1cm apart are given different colours. Can you prove that this is impossible if you have only three colours, but possible if you have seven?

The towers of Hanoi — Can you prove that this famous puzzle always has a solution, no matter how many pieces are involved?

Monochromatic cows — Here's a proof that all cows in a field are of the same colour. Can you find the flaw?

A proof with a hole — Here's a proof that π is equal to 2. Can you find the flaw?

## Elusive proofs

### The Riemann hypothesis

The Riemann hypothesis is one of the most famous problems in mathematics, whose proof has eluded mathematicians for 150 years. Here is a list of articles exploring it in detail:

### Fermat's last theorem

Fermat's last theorem — Fermat's last theorem had foxed mathematicians for over 400 years, when Andrew Wiles famously announced a proof in 1993. This collection of article explores the theorem, its proof, and some related topics.

### The Poincaré conjecture

The 100-year-old Poincaré conjecture made the headlines in 2006, unusually for a pure maths result. The reason was not so much the conjecture itself, but the strange character of the person who finally cracked it — Grigory Perelman even refused the substantial prize associated to proving the conjecture. The following articles explore the conjecture and track its exciting history:

- Proof of Poincaré?;
- Mathematical millionaire?;
- Exotic spheres, or why 4-dimensional space is a crazy place.
- The Fields Medals 2006.

### Harmless looking results about primes

The primes are notorious for posing easy-looking questions that turn out to be fiendishly difficult to answer. The following articles look at some of these:

- Maths in a minute: Number mysteries;
- Mathematical mysteries: the Goldbach conjecture;
- Goldbach revisited;
- Mathematical mysteries: twin primes;
- Mind the gap;
- Elusive twins.
- Primes without 7s;
- Twins move within reach.

### Bathroom secrets

The trouble with five — Squares do it, triangles do it, even hexagons do it — but pentagons don't. They just won't fit together to tile a flat surface. So are there any tilings based on fiveness? This article explores the unsolved five-fold tiling problem.

### Million dollar problems

Some of the problems mentioned above are included in the set of the Clay Institute's *Millennium Prize Problems*: anyone solving them will earn one million dollars. The following articles explore this list of problems, which rank among the world's most difficult and important:

- How maths can make you rich and famous;
- How maths can make you rich and famous: Part II;
- Code-breakers, doughnuts, and violins.

## Philosophical proofs

### Human versus machine

Can a computer be trusted to prove a theorem? It's a question that divides the mathematical world. The following articles probe the arguments, and look at famous results that have been proved by computers, including the four colour theorem and Kepler's conjecture:

- The future of proof;
- Will machine learning replace mathematicians?;
- Welcome to the maths lab;
- Country road, take me home;
- Automated maths.

### Infinite possibilities

What's the nature of infinity? Are all infinities the same? And what happens if you've got infinitely many infinities? The following articles explore how these questions gave rise to controversial maths which brought triumph to one man and ruin to another:

- A glimpse of Cantor's paradise;
- Cantor and Cohen: Infinite investigators part I;
- Cantor and Cohen: Infinite investigators part II.

### Constructive maths

Something from nothing? — If you can prove that a statement can't possibly be false, does this mean it's true? This article explores the law of the excluded middle and how it is used in mathematics.

Can maths exist if you can't see it? — Unlike physicists, mathematicians don't need to "see" something to be certain it exists. But are non-constructive proofs really valid? And what happens if you try and make do without them? This collection of articles explores a school of thought called constructivist mathematics.

### The limits of mathematics

One of the most ground-breaking results in mathematics is known as *Gödel's incompleteness theorem*, which, loosely-speaking, states that there are limits to what can be proved mathematically. The following articles look at the theorem, its history and its legacy: