INSEAD Library

Includes bibliographical references and index

Digitized

Proofs From The Book Table of Contents Number Theory 1 1. Six proofs of the infinity of primes ....................................................... 3 2. Bertrand's postulate ................................................................................ 7 3. Binomial coefficients are (almost) never powers ............................... 13 4. Representing numbers as sums of two squares ................................... 17 5. Every finite division ring is a field ...................................................... 23 6. Some irrational numbers ..................................................................... 27 7. Three times 7r2/ 6 ................................................................................... 35 Geometry 43 8. Hilbert's third problem: decomposing polyhedra ................................ 45 9. Lines in the plane and decompositions of graphs ............................... 53 10. The slope problem .............................................................................. 59 11. Three applications of Euler's formula ............................................... 65 12. Cauchy's rigidity theorem ................................................................. 71 13. Touching simplices ............................................................................ 75 14. Every large point set has an obtuse angle ......................................... 79 15. Borsuk's conjecture ........................................................................... 85 Analysis 91 16. Sets, functions, and the continuum hypothesis .................................. 93 17. In praise of inequalities ................................................................... 109 18. A theorem of Pólya on polynomials ................................................ 117 19. On a lemma of Littlewood and Offord ............................................ 123 20. Cotangent and the Herglotz trick .................................................... 127 21. Buffon's needle problem ................................................................. 133 Combinatorics 137 22. Pigeon-hole and double counting ..................................................... 139 23. Three famous theorems on finite sets .............................................. 151 24. Shuffling cards .................................................................................. 157 25. Lattice paths and determinants .......................................................... 167 26. Cayley's formula for the number of trees ......................................... 173 27. Completing Latin squares ................................................................. 179 28. The Dinitz problem .......................................................................... 185 29. Identities versus bijections ............................................................... 191 Graph Theory 197 30. Five-coloring plane graphs ............................................................... 199 31. How to guard a museum ................................................................... 203 32. Turán's graph theorem ...................................................................... 207 33. Communicating without errors ........................................................ 213 34. Of friends and politicians ................................................................. 223 35. Probability makes counting (sometimes) easy ................................. 227 About the Illustrations Index 236 237