NumberTheory 1
1．Sixproofs of the infinity of primes 3
2．Bertrand'spostulate 7
3．Binomial coefficients are(almost)never powers 13
4．Representing numbers as sums of two squares 17
5．The law of quadratic reciprocity 23
6．Every finite division ringis a field 31
7．Some irrational numbers 35
8．Three times π2/6 43
Geometry 51
9．Hilbert's third problem：decomposing polyhedra 53
10．Lines in the plane and decompositions of graphs 63
11．The slope problem 69
12．Three applications of Euler's formula 75
13．Cauchy's rigidity theorem 81
14．Touching simplices 85
15．Every large point set has an obtuse angle 89
16．Borsuk's conjecture 95
Analysis 101
17．Sets,functions,and the continuum hypothesis 103
18．Inpraise of inequalities 119
19．The fundamental theorem of algebra 127
20．One square and an odd number of triangles 131
21．Atheoremof 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
Combinatorics 159
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
Graph Theory 219
33．The Dinitz problem 221
34．Five-coloring plane graphs 227
35．How to guard a museum 231
36．Turán's graph theorem 235
37．Communicating without errors 241
38．The chromatic number of Kneser graphs 251
39．Of friends and politicians 257
40．Probability makes counting(sometimes)easy 261
About the Illustrations 270
Index 271