The Art of Mathematics: Coffee Time in MemphisCambridge University Press, 14.09.2006 - 359 Seiten Can a Christian escape from a lion? How quickly can a rumour spread? Can you fool an airline into accepting oversize baggage? Recreational mathematics is full of frivolous questions in which the mathematician's art can be brought to bear. But play often has a purpose, whether it's bear cubs in mock fights, or war games. In mathematics, it can sharpen skills, or provide amusement, or simply surprise, and collections of problems have been the stock-in-trade of mathematicians for centuries. Two of the twentieth century's greatest players of problem posing and solving, Erdos and Littlewood, are the inspiration for this collection, which is designed to be sipped from, rather than consumed, in one sitting. The questions themselves range in difficulty: the most challenging offer a glimpse of deep results that engage mathematicians today; even the easiest are capable of prompting readers to think about mathematics. All come with solutions, many with hints, and most with illustrations. Whether you are an expert, or a beginner, oran amateur, this book will delight for a lifetime. - Publisher. |
Inhalt
The Problems | 1 |
25 | 5 |
Contents ix | 13 |
86 | 20 |
888 | 28 |
The Hints | 36 |
The Solutions | 45 |
Erdős Problems for Epsilons | 48 |
Perfect Trees | 218 |
Circular sequences | 220 |
Infinite Sets with Integral Distances | 222 |
Finite Sets with Integral Distances | 223 |
Thues Theorem | 224 |
the ThueMorse Theorem | 226 |
the CauchyDavenport Theorem | 229 |
the ErdősGinzburgZiv Theorem | 232 |
Points on a Circle | 50 |
Partitions into Closed Sets | 52 |
Triangles and Squares | 53 |
Polygons and Rectangles | 55 |
African Rally | 56 |
Fixing Convex Domains | 58 |
30 | 59 |
Nested Subsets | 61 |
Almost Disjoint Subsets | 63 |
Loaded Dice | 64 |
An Unexpected Inequality | 65 |
the ErdősSelfridge Theorem | 66 |
Independent Sets | 68 |
Expansion into Sums 23 | 69 |
A Tennis Match | 70 |
Another Erdős Problem for Epsilons | 71 |
Planar Domains of Diameter 1 | 73 |
Orienting Graphs | 74 |
A Simple Clock | 75 |
Neighbours in a Matrix | 76 |
Separately Continuous Functions | 77 |
Boundary Cubes | 78 |
Lozenge Tilings | 79 |
31 | 93 |
34 | 103 |
36 | 110 |
40 | 116 |
43 | 124 |
46 | 133 |
49 | 140 |
52 | 147 |
55 | 153 |
57 | 159 |
61 | 165 |
Zagiers Inequality | 169 |
Squares Touching a Square | 170 |
Infection with Three Neighbours | 171 |
The Spread of Infection on a Torus | 173 |
Dominating Sequences | 174 |
Sums of Reciprocals | 175 |
Absentminded Passengers | 176 |
Airline Luggage | 177 |
the ErdősKoRado Theorem | 179 |
the MYBL Inequality | 180 |
Five Points in Space | 183 |
Triads | 184 |
Colouring Complete Graphs | 186 |
a Theorem of Besicovitch | 187 |
Independent Random Variables | 190 |
Triangles Touching a Triangle | 192 |
Even and Odd Graphs | 193 |
the MoonMoser Theorem | 194 |
Filling a Matrix | 197 |
the ErdősMordell Theorem | 199 |
Perfect Difference Sets | 203 |
Difference Bases | 205 |
the HardyLittlewood Maximal Theorem | 208 |
Random Words | 212 |
Crossing a Chess Board | 214 |
Powers of Paths and Cycles | 216 |
Powers of Oriented Cycles | 217 |
Subwords of Distinct Words | 237 |
Prime Factors of Sums | 238 |
Catalan Numbers | 240 |
Permutations without Long Decreasing Subsequences | 242 |
a Theorem of Justicz Scheinerman and Winkler | 244 |
the BrunnMinkowski Inequality | 246 |
Bollobáss Lemma | 248 |
Saturated Hypergraphs | 252 |
Hardys Inequality | 253 |
Carlemans Inequality | 257 |
Triangulating Squares | 259 |
Strongly Separating Families | 262 |
Strongly Separating Systems of Pairs of Sets | 263 |
The Maximum EdgeBoundary of a Downset | 265 |
Partitioning a Subset of the Cube | 267 |
Frankls Theorem | 269 |
Even Sets with Even Intersections | 271 |
Sets with Even Intersections | 273 |
Even Clubs | 275 |
Covering the Sphere | 276 |
Lovászs Theorem | 277 |
Partitions into Bricks | 279 |
Drawing Dense Graphs | 280 |
Székelys Theorem | 282 |
PointLine Incidences | 284 |
Geometric Graphs without Parallel Edges | 285 |
Shortest Tours | 288 |
Density of Integers | 291 |
Kirchbergers Theorem | 293 |
Chords of Convex Bodies | 294 |
Neighourly Polyhedra | 296 |
Perles Theorem | 299 |
The Rank of a Matrix | 301 |
a Theorem of Frankl and Wilson | 303 |
Families without Orthogonal Vectors | 306 |
the KahnKalai Theorem | 308 |
Periodic Sequences | 311 |
the FineWilf Theorem | 313 |
Wendels Theorem | 315 |
Planar and Spherical Triangles | 318 |
Hadžiivanovs theorem | 319 |
A Probabilistic Inequality | 321 |
Cube Slicing | 322 |
the HobbyRice Theorem | 324 |
Cutting a Necklace | 326 |
the RieszThorin Interpolation Theorem | 328 |
Uniform Covers | 332 |
Projections of Bodies | 333 |
the Box Theorem of Bollobás and Thomason | 335 |
the RayChaudhuri Wilson Inequality | 337 |
the FranklWilson Inequality | 340 |
Maps from Sn | 343 |
Hopfs Theorem | 344 |
Spherical Pairs | 345 |
Realizing Distances | 346 |
A Closed Cover of S² | 348 |
the Friendship Theorem of Erdős Rényi and Sós | 349 |
Polarities in Projective Planes | 352 |
Steinitzs Theorem | 353 |
The PointLine Game | 356 |
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a₁ Amer angle antipodal points assertion assume b₁ bipartite graphs Bollobás Borsuk's Borsuk's conjecture calls circle claimed closed sets colour combinatorial Combinatorics complete graph completing the proof conjecture Consequently contains contradiction convex sets cross-intersecting cube disjoint distance down-set edges Figure finite set function G.H. Hardy geometric graph Helly's theorem Hence hexagon hypergraph implies inequality infected sites integer intersection k-subsets least length Mathematics matrix maximal minimal number modulo n-dimensional natural numbers neighbours non-empty Notes number of elements odd number partition Paul Erdős plane polygonal polynomial prime primitive fixing problem prove random rectangles result segment sequence sets of diameter Show side-length simple tiling solution Sperner family subgraph subsets subtree summands Suppose Sylvester-Gallai theorem theorem tree triangle trivial unit square vectors vertex vertex set vertices write