Sinopsis
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. Touching simplices.- 13. Every large point set has an obtuse angle.- 14. Borsuk's conjecture.- Analysis.- 15. Sets, functions, and the continuum hypothesis.- 16. In praise of inequalities.- 17. A theorem of Pólya on polynomials.- 18. On a lemma of Littlewood and Offord.- 19. Cotangent and the Herglotz trick.- 20. Buffon's needle problem.- Combinatorics.- 21. Pigeon-hole and double counting.- 22. Three famous theorems on finite sets.- 23. Lattice paths and determinants.- 24. Cayley's formula for the number of trees.- 25. Completing Latin squares.- 26. The Dinitz problem.- Graph Theory.- 27. Five-coloring plane graphs.- 28. How to guard a museum.- 29. Turán's graph theorem.- 30. Communicating without errors.- 31. Of friends and politicians.- 32. Probability makes counting (sometimes) easy.- About the Illustrations.
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