The Art of Mathematics: Coffee Time in Memphis

The Art of Mathematics: Coffee Time in Memphis

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by Bela Bollobas
     
 

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… See more details below

Overview

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.

About the Author:
Bela Bollobas is a Senior Research Fellow at Trinity College, Cambridge and holds the jabie Hardin Chair of Excellence in Combinatorics at the University of Memphis

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Product Details

ISBN-13:
9780521693950
Publisher:
Cambridge University Press
Publication date:
09/28/2006
Pages:
376
Product dimensions:
5.98(w) x 8.98(h) x 0.87(d)

Table of Contents


Preface     xiii
The Problems     1
The Hints     36
The Solutions     45
The Lion and The Christian     45
Integer Sequences: Erdos Problems for Epsilons     48
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
Nested Subsets     61
Almost Disjoint Subsets     63
Loaded Dice     64
An Unexpected Inequality     65
Colouring Lines: the Erdos-Selfridge Theorem     66
Independent Sets     68
Expansion into Sums 2i 3j     69
A Tennis Match     70
A Triangle Inequality: 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
Convexity and Intersecting Simplices: the Theorems of Radon and Caratheodory     88
Intersecting Convex Sets: Helly's Theorem     90
Judicious Partitions of Points     92
Further Lozenge Tilings     93
Two Squares in a Square     95
Lines Through Points: the Sylvester-Gallai Theorem     98
The Spread of Infection on a Square Grid     104
The Spread of Infection in a d-dimensional Box     106
Sums of Integers: an Easy Erdos Problem for Epsilons     110
Normal Numbers: the Champernowne Number     111
Random Walks on Graphs     113
Simple Tilings of Rectangles     114
L-tilings     116
Antipodal Points and Maps: Borsuk's Theorem     117
Bodies of Diameter 1: Borsuk's Problem     120
Equilateral Triangles: Napoleon's Theorem     124
Trisectors of Angles: Morley's Theorem     126
Connected Subgraphs     129
Subtrees of an Infinite Tree     133
Two-distance Sets     134
Gossiping Dons     136
Exact Covers: the de Bruijn-Erdos Theorem     140
Constant Intersections: an Extension of the de Bruijn-Erdos 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
A Converse of Cauchy's Inequality: Zagier's 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
Absent-minded Passengers     176
Airline Luggage     177
Intersecting Sets: the Erdos-Ko-Rado Theorem     179
Sperner Families: the MYBL Inequality     180
Five Points in Space     183
Triads     184
Colouring Complete Graphs     186
Symmetric Convex Domains: a Theorem of Besicovitch     187
Independent Random Variables     190
Triangles Touching a Triangle      192
Even and Odd Graphs     193
Packing Squares: the Moon-Moser Theorem     194
Filling a Matrix     197
An Inequality Concerning Triangles: the Erdos-Mordell Theorem     199
Perfect Difference Sets     203
Difference Bases     205
Satisfied Cricketers: the Hardy-Littlewood Maximal Theorem     208
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
Cube-free Words: Thue's Theorem     224
Square-free Words: the Thue-Morse Theorem     226
Addition of Residue Classes: the Cauchy-Davenport Theorem     229
Sums Congruent to Zero: the Erdos-Ginzburg-Ziv Theorem     232
Subwords of Distinct Words     237
Prime Factors of Sums     238
Catalan Numbers     240
Permutations without Long Decreasing Subsequences     242
Random Intervals: a Theorem of Justicz, Scheinerman and Winkler     244
Sums of Convex Bodies: the Brunn-Minkowski Inequality     246
Cross-Intersecting Families: Bollobas's Lemma     248
Saturated Hypergraphs     252
The Norm of Averages: Hardy's Inequality     253
The Average of Geometric Means: Carleman's Inequality     257
Triangulating Squares     259
Strongly Separating Families     262
Strongly Separating Systems of Pairs of Sets     263
The Maximum Edge-Boundary of a Down-set     265
Partitioning a Subset of the Cube     267
Weakly Cross-intersecting Pairs: Frankl's Theorem     269
Even Sets with Even Intersections     271
Sets with Even Intersections     273
Even Clubs     275
Covering the Sphere     276
The Kneser Graph: Lovasz's Theorem     277
Partitions into Bricks     279
Drawing Dense Graphs     280
Unit Distances: Szekely's Theorem     282
Point-Line Incidences     284
Geometric Graphs without Parallel Edges     285
Shortest Tours     288
Density of Integers     291
Black and White Sheep: Kirchberger's Theorem     293
Chords of Convex Bodies     294
Neighourly Polyhedra      296
Neighbourly Simplices: Perles' Theorem     299
The Rank of a Matrix     301
Modular Intersecting k-uniform Set Systems: a Theorem of Frankl and Wilson     303
Families without Orthogonal Vectors     306
A Counterexample to Borsuk's Conjecture: the Kahn-Kalai Theorem     308
Periodic Sequences     311
Periodic Words: the Fine-Wilf Theorem     313
Points on a Hemisphere: Wendel's Theorem     315
Planar and Spherical Triangles     318
Hobnails: Hadziivanov's theorem     319
A Probabilistic Inequality     321
Cube Slicing     322
Measures on [0, 1]: the Hobby-Rice Theorem     324
Cutting a Necklace     326
The Norm of an Operator: the Riesz-Thorin Interpolation Theorem     328
Uniform Covers     332
Projections of Bodies     333
BTBT: the Box Theorem of Bollobas and Thomason     335
Intersecting Uniform Set Systems: the Ray-Chaudhuri-Wilson Inequality     337
Intersecting Set Systems: the Frankl-Wilson Inequality     340
Maps from S[superscript n]     342
Closed Covers of S[superscript n]: Hopf's Theorem     344
Spherical Pairs     345
Realizing Distances     345
A Closed Cover of S[superscript 2]     348
A Friendly Party: the Friendship Theorem of Erdos, Renyi and Sos     349
Polarities in Projective Planes     352
Permutations of Vectors: Steinitz's Theorem     353
The Point-Line Game     356

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