# Proofs from THE BOOK

### Number Theory

1. Six proofs of the infinity of primes
2. Bertrand's postulate
3. Binomial coefficients are (almost) never powers
4. Representing numbers as sums of two squares
5. Every finite division ring is a field
6. Some irrational numbers

### Geometry

7. Hilbert's third problem: decomposing polyhedra
8. Lines in the plane and decompositions of graphs
9. The slope problem
10. Three applications of Euler's formula
11. Cauchy's rigidity theorem
12. The problem of the thirteen spheres
13. Touching simplices
14. Every large point set has an obtuse angle
15. Borsuk's conjecture

### Analysis

16. Sets, functions, and the continuum hypothesis
17. In praise of inequalities
18. A theorem of Pólya on polynomials
19. On a lemma of Littlewood and Offord

### Combinatorics

20. Pigeon-hole and double counting
21. Three famous theorems on finite sets
22. Cayley's formula for the number of trees
23. Completing Latin squares
24. The Dinitz problem

### Graph Theory

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