Table of Contents

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 π2/6.- 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 Pólya on polynomials.- 24. On a lemma of Littlewood and Offord.- 25. Cotangent and the Herglotz trick.- 26. Buffon’s needle problem.- Combinatorics: 27. Pigeon-hole and double counting.- 28. Tiling rectangles.- 29. Three famous theorems on finite sets.- 30. Shuffling cards.- 31. Lattice paths and determinants.- 32. Cayley’s formula for the number of trees.- 33. Identities versus bijections.- 34. The finite Kakeya problem.- 35. Completing Latin squares.- Graph Theory: 36. The Dinitz problem.- 37. Permanents and the po

wer of entropy.- 38. Five-coloring plane graphs.- 39. How to guard a museum.- 40. Turán’s graph theorem.- 41. Communicating without errors.- 42. The chromatic number of Kneser graphs.- 43. Of friends and politicians.- 44. Probability makes counting (sometimes) easy.- About the Illustrations.- Index.

About the Author

Martin Aigner received his Ph.D. from the University of Vienna and has been professor of mathematics at the Freie Universität Berlin since 1974. He has published in various fields of combinatorics and graph theory and is the author of several monographs on discrete mathematics, among them the Springer books Combinatorial Theory and A Course on Enumeration. Martin Aigner is a recipient of the 1996 Lester R. Ford Award for mathematical exposition of the Mathematical Association of America MAA.

Günter M. Ziegler received his Ph.D. from M.I.T. and has been professor of mathematics in Berlin – first at TU Berlin, now at Freie Universität – since 1995. He has published in discrete mathematics, geometry, topology, and optimization, including the Lectures on Polytopes with Springer, as well as „Do I Count? Stories from Mathematics“. Günter M. Ziegler is a recipient of the 2006 Chauvenet Prize of the MAA for his expository writing and the 2008 Communicator award of the German Science Foundation.

Martin Aigner and Günter M. Ziegler have started their work on Proofs from THE BOOK in 1995 together with Paul Erdös. The first edition of this book appeared in 1998 – it has since been translated into 13 languages: Brazilian, Chinese, German, Farsi, French, Hungarian, Italian, Japanese, Korean, Polish, Russian, Spanish, and Turkish.

Reviews

“This book by Aigner and Ziegler, now in its fifth edition, seeks to pay homage to the late Paul Erdõs by attempting to provide an approximation of `The Book.’ … Throughout, illustrations and figures are used to support the arguments in the main text; these can greatly help the readability of the proofs, especially for novices like me. … the book is a marvelous project and this new edition provides a good amount of fresh material.” (Harry Strange, Computing Reviews, March, 2015)