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Analysis 17 Sets, functions, and the continuum hypothesis 103 18 In praise of inequalities 119 19 The fundamental theorem of algebra 127 20 One square and an odd number of triangles 131 21 A theorem of Pólya on polynomials 139 22 On a lemma of Littlewood and Offord 145 23 Cotangent and the Herglotz trick 149 24 Buffon’s needle problem 155 “Hilbert’s seaside resort hotel”
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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
Page 59 and 60: 50 Three times π 2 /6 (3) It has b
Page 62 and 63: Hilbert’s third problem: decompos
Page 70: Hilbert’s third problem: decompos
Page 73 and 74: 64 Lines in the plane, and decompos
Page 75 and 76: 66 Lines in the plane, and decompos
Page 78 and 79: The slope problem Chapter 11 Try fo
Page 80 and 81: The slope problem 71 • Every move
Page 82: The slope problem 73 For the second
Page 85 and 86: 76 Three applications of Euler’s
Page 91 and 92: 82 Cauchy’s rigidity theorem Q: Q
Page 93 and 94: 84 Cauchy’s rigidity theorem A be
Page 95 and 96: 86 Touching simplices l l l ′ Pr
Page 97 and 98: 88 Touching simplices For our examp
Page 99 and 100: 90 Every large point set has an obt
Page 105 and 106: 96 Borsuk’s conjecture Theorem. L
Page 107 and 108: 98 Borsuk’s conjecture On the oth
Page 109: 100 Borsuk’s conjecture To obtain
Page 113 and 114: 104 Sets, functions, and the contin
Page 129 and 130: 120 In praise of inequalities For n
Page 131 and 132: 122 In praise of inequalities We fi
Page 133 and 134: 124 In praise of inequalities Apply
Page 136 and 137: The fundamental theorem of algebra
Page 138: The fundamental theorem of algebra
Page 141 and 142: 132 One square and an odd number of
Page 149 and 150: 140 A theorem of Pólya on polynomi
Page 155 and 156: 146 On a lemma of Littlewood and Of
Page 157 and 158: 148 On a lemma of Littlewood and Of
Page 159 and 160: 150 Cotangent and the Herglotz tric
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152 Cotangent and the Herglotz tric
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154 Cotangent and the Herglotz tric
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156 Buffon’s needle problem If yo
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158 Buffon’s needle problem Refer
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Pigeon-hole and double counting Cha
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Pigeon-hole and double counting 163
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174 Tiling rectangles Since ∫ d
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176 Tiling rectangles Next comes a
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Three famous theorems on finite set
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186 Shuffling cards which is smalle
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188 Shuffling cards For card player
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190 Shuffling cards and we have see
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192 Shuffling cards Bell Labs “Ma
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194 Shuffling cards k d(k) 1 1.000
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196 Lattice paths and determinants
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202 Cayley’s formula for the numb
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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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