# Proofs from THE BOOK

## Table of Contents

### Number Theory

- Six proofs of the infinity of primes
- Bertrand's postulate
- Binomial coefficients are (almost) never powers
- Representing numbers as sums of two squares
- Every finite division ring is a field
- Some irrational numbers
### Geometry

- Hilbert's third problem: decomposing polyhedra
- Lines in the plane and decompositions of graphs
- The slope problem
- Three applications of Euler's formula
- Cauchy's rigidity theorem
- The problem of the thirteen spheres
- Touching simplices
- Every large point set has an obtuse angle
- Borsuk's conjecture
### Analysis

- Sets, functions, and the continuum hypothesis
- In praise of inequalities
- A theorem of Pólya on polynomials
- On a lemma of Littlewood and Offord
### Combinatorics

- Pigeon-hole and double counting
- Three famous theorems on finite sets
- Cayley's formula for the number of trees
- Completing Latin squares
- The Dinitz problem
### Graph Theory

- Five-coloring plane graphs
- How to guard a museum
- Turán's graph theorem
- Communicating without errors
- Of friends and politicians
- Probability makes counting (sometimes) easy

### About the Illustrations

### Index

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Copyright © 1996-2018 Alexander Bogomolny

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