Proofs from THE BOOK This revised and enlarged sixth edition of Proofs from THE BOOK features an entirely new chapter on Van der Waerden’s permanent conjecture as well as additional highly original and delightful proofs in other chapters. From the citation on the occasion of the 2018 Steele Prize for Mathematical Exposition “… It is almost impossible to write a mathematics book that can be read and enjoyed by people of all levels and backgrounds yet Aigner and Ziegler accomplish this feat of exposition with virtuoso style. […] This book does an invaluable service to mathematics by illustrating for non mathematicians what it is that mathematicians mean when they speak about beauty.” From the Reviews .. Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages one gem after another. .. Aigner and Ziegler.. write: .. all we offer is the examples that we have selected hoping that our readers will share our enthusiasm about brilliant ideas clever insights and wonderful observations. I do. .. Notices of the AMS August 1999 .. This book is a pleasure to hold and to look at: ample margins nice photos instructive pictures and beautiful drawings .. It is a pleasure to read as well: the style is clear and entertaining the level is close to elementary the necessary background is given separately and the proofs are brilliant. .. LMS Newsletter January 1999 Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental their proofs so elegant and the remaining open questions so intriguing that every mathematician regardless of speciality can benefit from reading this book. .. SIGACT News December 2011 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. Van der Waerdens 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. Turán’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