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Preface Paul Erdős liked to talk about The Book, in which God maintains the perfect <strong>proofs</strong> for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdős also said that you need not believe in God but, as a mathematician, you should believe in The Book. A few years ago, we suggested to him to write up a first (and very modest) approximation to The Book. He was enthusiastic about the idea and, characteristically, went to work immediately, filling page after page with his suggestions. Our book was supposed to appear in March 1998 as a present to Erdős’ 85th birthday. With Paul’s unfortunate death in the summer of 1996, he is not listed as a co-author. Instead this book is dedicated to his memory. We have no definition or characterization of what constitutes a proof from The Book: all we offer here is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations. We also hope that our readers will enjoy this despite the imperfections of our exposition. The selection is to a great extent influenced by Paul Erdős himself. A large number of the topics were suggested by him, and many of the <strong>proofs</strong> trace directly back to him, or were initiated by his supreme insight in asking the right question or in making the right conjecture. So to a large extent this book reflects the views of Paul Erdős as to what should be considered a proof from The Book. A limiting factor for our selection of topics was that everything in this book is supposed to be accessible to readers whose backgrounds include only a modest amount of technique from undergraduate mathematics. A little linear algebra, some basic analysis and number theory, and a healthy dollop of elementary concepts and reasonings from discrete mathematics should be sufficient to understand and enjoy everything in this book. We are extremely grateful to the many people who helped and supported us with this project — among them the students of a seminar where we discussed a preliminary version, to Benno Artmann, Stephan Brandt, Stefan Felsner, Eli Goodman, Torsten Heldmann, and Hans Mielke. We thank Margrit Barrett, Christian Bressler, Ewgenij Gawrilow, Michael Joswig, Elke Pose, and Jörg Rambau for their technical help in composing this book. We are in great debt to Tom Trotter who read the manuscript from first to last page, to Karl H. Hofmann for his wonderful drawings, and most of all to the late great Paul Erdős himself. Paul Erdős “The Book” Berlin, March 1998 Martin Aigner · Günter M. Ziegler
VI Preface to the Fourth Edition When we started this project almost fifteen years ago we could not possibly imagine what a wonderful and lasting response our book about The Book would have, with all the warm letters, interesting comments, new editions, and as of now thirteen translations. It is no exaggeration to say that it has become a part of our lives. In addition to numerous improvements, partly suggested by our readers, the present fourth edition contains five new chapters: Two classics, the law of quadratic reciprocity and the fundamental theorem of algebra, two chapters on tiling problems and their intriguing solutions, and a highlight in graph theory, the chromatic number of Kneser graphs. We thank everyone who helped and encouraged us over all these years: For the second edition this included Stephan Brandt, Christian Elsholtz, Jürgen Elstrodt, Daniel Grieser, Roger Heath-Brown, Lee L. Keener, Christian Lebœuf, Hanfried Lenz, Nicolas Puech, John Scholes, Bernulf Weißbach, and many others. The third edition benefitted especially from input by David Bevan, Anders Björner, Dietrich Braess, John Cosgrave, Hubert Kalf, Günter Pickert, Alistair Sinclair, and Herb Wilf. For the present edition, we are particularly grateful to contributions by France Dacar, Oliver Deiser, Anton Dochtermann, Michael Harbeck, Stefan Hougardy, Hendrik W. Lenstra, Günter Rote, Moritz Schmitt, and Carsten Schultz. Moreover, we thank Ruth Allewelt at Springer in Heidelberg as well as Christoph Eyrich, Torsten Heldmann, and Elke Pose in Berlin for their help and support throughout these years. And finally, this book would certainly not look the same without the original design suggested by Karl-Friedrich Koch, and the superb new drawings provided for each edition by Karl H. Hofmann. Berlin, July 2009 Martin Aigner · Günter M. Ziegler
Page 2: Martin Aigner Günter M. Ziegler Pr
Page 5: Prof. Dr. Martin Aigner FB Mathemat
Page 9 and 10: VIII Table of Contents 21. A theore
Page 12 and 13: Six proofs of the infinity of prime
Page 14 and 15: Six proofs of the infinity of prime
Page 16 and 17: Bertrand’s postulate Chapter 2 We
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Page 22 and 23: Binomial coefficients are (almost)
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Page 26 and 27: Representing numbers as sums of two
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Page 40 and 41: Every finite division ring is a fie
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Page 44 and 45: Some irrational numbers Chapter 7
Page 46 and 47: Some irrational numbers 37 and this
Page 48 and 49: Some irrational numbers 39 F(x) may
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Page 55 and 56: 46 Three times π 2 /6 For m = 1, 2
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48 Three times π 2 /6 1 f(t) = 1 t
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50 Three times π 2 /6 (3) It has b
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Hilbert’s third problem: decompos
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64 Lines in the plane, and decompos
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66 Lines in the plane, and decompos
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The slope problem Chapter 11 Try fo
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The slope problem 71 • Every move
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The slope problem 73 For the second
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76 Three applications of Euler’s
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82 Cauchy’s rigidity theorem Q: Q
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84 Cauchy’s rigidity theorem A be
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86 Touching simplices l l l ′ Pr
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88 Touching simplices For our examp
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90 Every large point set has an obt
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96 Borsuk’s conjecture Theorem. L
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98 Borsuk’s conjecture On the oth
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100 Borsuk’s conjecture To obtain
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Sets, functions, and the continuum
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In praise of inequalities Chapter 1
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In praise of inequalities 121 where
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In praise of inequalities 123 Proo
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In praise of inequalities 125 Seco
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128 The fundamental theorem of alge
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One square and an odd number of tri
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A theorem of Pólya on polynomials
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On a lemma of Littlewood and Offord
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On a lemma of Littlewood and Offord
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Cotangent and the Herglotz trick Ch
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Cotangent and the Herglotz trick 15
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Cotangent and the Herglotz trick 15
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Buffon’s needle problem Chapter 2
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Buffon’s needle problem 157 The c
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Combinatorics 25 Pigeon-hole and do
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162 Pigeon-hole and double counting
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Tiling rectangles Chapter 26 Some m
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Tiling rectangles 175 Third proof.
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Tiling rectangles 177 References [1
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180 Three famous theorems on finite
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182 Three famous theorems on finite
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Shuffling cards Chapter 28 How ofte
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Shuffling cards 187 Indeed, if A i
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Shuffling cards 189 Let T be the nu
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Shuffling cards 191 Theorem 1. Let
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Shuffling cards 193 Proof. (1) We
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Lattice paths and determinants Chap
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Lattice paths and determinants 197
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Lattice paths and determinants 199
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Cayley’s formula for the number o
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Identities versus bijections Chapte
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Identities versus bijections 209 Pr
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Identities versus bijections 211 As
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Completing Latin squares Chapter 32
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Completing Latin squares 215 Proof
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Completing Latin squares 217 be dis
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Graph Theory 33 The Dinitz problem
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222 The Dinitz problem In 1976 Vizi
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224 The Dinitz problem X A bipartit
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226 The Dinitz problem G : L(G) : a
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228 Five-coloring plane graphs From
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230 Five-coloring plane graphs is s
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232 How to guard a museum The follo
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234 How to guard a museum “Museum
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236 Turán’s graph theorem Let us
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238 Turán’s graph theorem Thus b
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Communicating without errors Chapte
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Communicating without errors 243 cl
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Communicating without errors 245 Le
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Communicating without errors 247 On
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Communicating without errors 249 No
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The chromatic number of Kneser grap
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258 Of friends and politicians w 1
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Probability makes counting (sometim
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Proofs from THE BOOK 269
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Index acyclic directed graph, 196 a
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Index 273 matrix-tree theorem, 203
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