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. The law of quadratic reciprocity
6. Every finite division ring is a field
7. The spectral theorem and Hadamard's determinant problem
8. Some irrational numbers
9. Three times [pi]2/6
Geometry: 10. Hilbert's third problem: decomposing polyhedral
11. Lines in the plane and decompositions of graphs
12. The slope problem
13. Three applications of Euler's formula
14. Cauchy's rigidity theorem
15. The Borromean rings don't exist
16. Touching simplices
17. Every large point set has an obtuse angle
18. Borsuk's conjecture
Analysis: 19. Sets, functions, and the continuum hypothesis
20. In praise of inequalities
21. The fundamental theorem of algebra
22. One square and an odd number of triangles
23. A theorem of P{acute}olya on polynomials
24. On a lemma of Littlewood and Offord
25. Cotangent and the Herglotz trick
26. Buffon's needle problem
Combinatorics: 27. Pigeon-hole and double counting
28. Tiling rectangles
29. Three famous theorems on finite sets
30. Shuffling cards
31. Lattice paths and determinants
32. Cayley's formula for the number of trees
33. Identities versus bijections
34. The finite Kakeya problem
35. Completing Latin squares
Graph Theory: 36. The Dinitz problem
37. Permanents and the power of entropy
38. Five-coloring plane graphs
39. How to guard a museum
40. Tur{acute}an's graph theorem
41. Communicating without errors
42. The chromatic number of Kneser graphs
43. Of friends and politicians
44. Probability makes counting (sometimes) easy
About the Illustrations
Index.