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