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
7. Three times π2/6
Geometry
8. Hilbert‘s third problem: decomposing polyhedra
9. Lines in the plane and decompositions of graphs
10. The slope problem
11. Three applications of Euler‘s formula
12. Cauchy‘s rigidity theorem
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
22. Pigeon-hole and double counting
23. Three famous theorems on finite sets
24. Shuffling cards
25. Lattice paths and determinants
26. Cayley's formulafor the number of trees
27. Completing Latin squares
28. The Dinitz problem
29. Identities versus bijections
Graph Theory
30. Five-coloring plane graphs
31. How to guard a museum
32. Turan's graph theorem
33. Communicating without e~ors
34. Of friends and pohtici~s
35. Probability makes counting (sometinles) easy
About the Illustrations
Index