Proofs from the Book (Hardcover, 6, 2018)

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 pi2/6.- 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 Polya on polynomials.- 24. Van der Waerden's permanent conjecture.- 25. On a lemma of Littlewood and Offord.- 26. Cotangent and the Herglotz trick.- 27. Buffon's needle problem.- Combinatorics: 28. Pigeon-hole and double counting.- 29. Tiling rectangles.- 30. Three famous theorems on finite sets.- 31. Shuffling cards.- 32. Lattice paths and determinants.- 33. Cayley's formula for the number of trees.- 34. Identities versus bijections.- 35. The finite Kakeya problem.- 36. Completing Latin squares.- Graph Theory: 37. Permanents and the power of entropy.- 38. The Dinitz problem.- 39. Five-coloring plane graphs.- 40. How to guard a museum.- 41. Turan's graph theorem.- 42. Communicating without errors.- 43. The chromatic number of Kneser graphs.- 44. Of friends and politicians.- 45. Probability makes counting (sometimes) easy.- About the Illustrations.- Index.