## The Art of Mathematics: Coffee Time in MemphisCan a Christian escape from a lion? How quickly can a rumor spread? Can you fool an airline into accepting oversize baggage? Recreational mathematics is full of frivolous questions where the mathematician's art can be brought to bear. But play often has a purpose. In mathematics, it can sharpen skills, provide amusement, or simply surprise, and books of problems have been the stock-in-trade of mathematicians for centuries. This collection is designed to be sipped from, rather than consumed in one sitting. The questions range in difficulty: the most challenging offer a glimpse of deep results that engage mathematicians today; even the easiest prompt readers to think about mathematics. All come with solutions, many with hints, and most with illustrations. Whether you are an expert, or a beginner or an amateur mathematician, this book will delight for a lifetime. |

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### Contents

The Problems | 1 |

Contents ix | 13 |

The Hints | 36 |

The Solutions | 45 |

Erdos Problems for Epsilons | 48 |

Points on a Circle | 50 |

Partitions into Closed Sets | 52 |

Triangles and Squares | 53 |

Triangles Touching a Triangle | 192 |

Even and Odd Graphs | 193 |

the MoonMoser Theorem | 194 |

Filling a Matrix | 197 |

the ErdosMordell Theorem | 199 |

Perfect Difference Sets | 203 |

Difference Bases | 205 |

the HardyLittlewood Maximal Theorem | 208 |

Polygons and Rectangles | 55 |

African Rally | 56 |

Fixing Convex Domains | 58 |

Nested Subsets | 61 |

Almost Disjoint Subsets | 63 |

Loaded Dice | 64 |

An Unexpected Inequality | 65 |

the ErdosSelfridge Theorem | 66 |

Independent Sets | 68 |

Expansion into Sums 2i3j | 69 |

A Tennis Match | 70 |

Another Erdos 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 |

A Continuum Independent Set | 83 |

Separating Families of Sets | 84 |

Bipartite Covers of Complete Graphs | 86 |

the Theorems | 88 |

of Radon and Caratheodory | 89 |

Hellys Theorem | 90 |

Judicious Partitions of Points | 92 |

Further Lozenge Tilings | 93 |

Two Squares in a Square | 95 |

the SylvesterGallai Theorem | 98 |

The Spread of Infection on a Square Grid | 104 |

The Spread of Infection in a ddimensional Box | 106 |

an Easy Erdos Problem for Epsilons | 110 |

the Champernowne Number | 111 |

Random Walks on Graphs | 113 |

Simple Tilings of Rectangles | 114 |

Ltilings | 116 |

Borsuks Theorem | 117 |

Borsuks Problem | 120 |

Napoleons Theorem | 124 |

Morleys Theorem | 126 |

Connected Subgraphs | 129 |

Subtrees of an Inﬁnite Tree | 133 |

Twodistance Sets | 134 |

Gossiping Dons | 136 |

the de BruijnErdos Theorem | 140 |

an Extension of the de BruijnErdos Theorem | 142 |

Bell Numbers | 144 |

Circles Touching a Square | 147 |

Gambling | 149 |

Complex Sequences | 151 |

Partitions of Integers | 153 |

Emptying Glasses | 157 |

Distances in Planar Sets | 159 |

Monic Polynomials | 161 |

Odd Clubs | 163 |

A Politically Correct Town | 164 |

Lattice Paths | 165 |

Triangulations of Polygons | 168 |

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 ErdosKoRado 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 |

Random Words | 212 |

Crossing a Chess Board | 214 |

Powers of Paths and Cycles | 216 |

Powers of Oriented Cycles | 217 |

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 ErdosGinzburgZiv Theorem | 232 |

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 |

Bollobass Lemma | 251 |

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 |

Lovaszs Theorem | 277 |

Partitions into Bricks | 279 |

Drawing Dense Graphs | 280 |

Szekelys 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 |

Hadˇziivanovs 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 Bollobas and Thomason | 335 |

the RayChaudhuri Wilson Inequality | 337 |

the FranklWilson Inequality | 340 |

Maps from Sn | 342 |

Hopfs Theorem | 344 |

Spherical Pairs | 345 |

Realizing Distances | 346 |

A Closed Cover of S2 | 348 |

the Friendship Theorem of Erdos Renyi and Sos | 349 |

Polarities in Projective Planes | 352 |

Steinitzs Theorem | 353 |

The PointLine Game | 356 |

### Common terms and phrases

Amer angle antipodal points assertion assume bipartite graphs Bollob´as Borsuks calls circle claimed closed sets colour combinatorial complete bipartite graphs complete graph completing the proof conjecture Consequently contains contradiction convex sets cube deﬁne disjoint distance down-set edges Erd˝os Figure ﬁnd ﬁnite set ﬁrst function G.H. Hardy Graph Theory Hence hexagon hypergraph hyperplane implies inequality infected sites integer intersection k-subsets least length London Math Mathematics matrix maximal minimal number modulo n-dimensional natural numbers neighbours non-empty Notes number of elements odd number partition plane polygonal polynomial problem prove random rectangles reﬂection result segment sequence sets of diameter Show side-length simple tiling solution Sperner family square-free subgraph subsets subtree sufﬁces summands Suppose SylvesterGallai theorem theorem triangle trivial unit square vectors vertex vertex set vertices write Young diagram