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Combinatorics 25 Pigeon-hole and double counting 161 26 Tiling rectangles 173 27 Three famous theorems on finite sets 179 28 Shuffling cards 185 29 Lattice paths and determinants 195 30 Cayley’s formula for the number of trees 201 31 Identities versus bijections 207 32 Completing Latin squares 213 “A melancholic Latin square”
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Martin Aigner Günter M. Ziegler Pr
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Prof. Dr. Martin Aigner FB Mathemat
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VI Preface to the Fourth Edition Wh
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VIII Table of Contents 21. A theore
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Six proofs of the infinity of prime
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Six proofs of the infinity of prime
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Bertrand’s postulate Chapter 2 We
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Bertrand’s postulate 9 times. Her
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Bertrand’s postulate 11 by compar
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Binomial coefficients are (almost)
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Binomial coefficients are (almost)
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Representing numbers as sums of two
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The law of quadratic reciprocity Ch
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The law of quadratic reciprocity 25
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Every finite division ring is a fie
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Every finite division ring is a fie
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Some irrational numbers Chapter 7
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Some irrational numbers 37 and this
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Some irrational numbers 39 F(x) may
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Some irrational numbers 41 Now assu
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44 Three times π 2 /6 v 1 1 2 y v
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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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Page 118 and 119: Sets, functions, and the continuum
Page 128 and 129: In praise of inequalities Chapter 1
Page 130 and 131: In praise of inequalities 121 where
Page 132 and 133: In praise of inequalities 123 Proo
Page 134: In praise of inequalities 125 Seco
Page 137 and 138: 128 The fundamental theorem of alge
Page 140 and 141: One square and an odd number of tri
Page 148 and 149: A theorem of Pólya on polynomials
Page 154 and 155: On a lemma of Littlewood and Offord
Page 156 and 157: On a lemma of Littlewood and Offord
Page 158 and 159: Cotangent and the Herglotz trick Ch
Page 160 and 161: Cotangent and the Herglotz trick 15
Page 162 and 163: Cotangent and the Herglotz trick 15
Page 164 and 165: Buffon’s needle problem Chapter 2
Page 166 and 167: Buffon’s needle problem 157 The c
Page 170 and 171: Pigeon-hole and double counting Cha
Page 172 and 173: Pigeon-hole and double counting 163
Page 180: Pigeon-hole and double counting 171
Page 183 and 184: 174 Tiling rectangles Since ∫ d
Page 185 and 186: 176 Tiling rectangles Next comes a
Page 188 and 189: Three famous theorems on finite set
Page 190 and 191: Three famous theorems on finite set
Page 192: Three famous theorems on finite set
Page 195 and 196: 186 Shuffling cards which is smalle
Page 197 and 198: 188 Shuffling cards For card player
Page 199 and 200: 190 Shuffling cards and we have see
Page 201 and 202: 192 Shuffling cards Bell Labs “Ma
Page 203 and 204: 194 Shuffling cards k d(k) 1 1.000
Page 205 and 206: 196 Lattice paths and determinants
Page 211 and 212: 202 Cayley’s formula for the numb
Page 217 and 218: 208 Identities versus bijections th
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210 Identities versus bijections Eu
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212 Identities versus bijections Re
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214 Completing Latin squares 1 2 3
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216 Completing Latin squares Inequa
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218 Completing Latin squares In our
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The Dinitz problem Chapter 33 The f
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The Dinitz problem 223 To state our
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The Dinitz problem 225 Claim. When
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Five-coloring plane graphs Chapter
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Five-coloring plane graphs 229 For
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How to guard a museum Chapter 35 He
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How to guard a museum 233 convex ve
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Turán’s graph theorem Chapter 36
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Turán’s graph theorem 237 all ve
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Turán’s graph theorem 239 Case 2
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242 Communicating without errors se
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244 Communicating without errors by
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246 Communicating without errors No
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248 Communicating without errors
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250 Communicating without errors Re
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252 The chromatic number of Kneser
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254 The chromatic number of Kneser
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Of friends and politicians Chapter
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Of friends and politicians 259 Now
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262 Probability makes counting (som
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About the Illustrations We are happ
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272 Index division ring, 31 double
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274 Index tangential rectangle, 121
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