Proofs from THE BOOK

Front Matter....Pages I-VIII
Front Matter....Pages 1-1
Six proofs of the infinity of primes....Pages 3-6
Bertrand’s postulate....Pages 7-12
Binomial coefficients are (almost) never powers....Pages 13-16
Representing numbers as sums of two squares....Pages 17-22
Every finite division ring is a field....Pages 23-26
Some irrational numbers....Pages 27-33
Three times π 2 /6....Pages 35-42
Front Matter....Pages 43-43
Hilbert’s third problem: decomposing polyhedra....Pages 45-51
Lines in the plane and decompositions of graphs....Pages 53-57
The slope problem....Pages 59-63
Three applications of Euler’s formula....Pages 65-70
Cauchy’s rigidity theorem....Pages 71-74
Touching simplices....Pages 75-78
Every large point set has an obtuse angle....Pages 79-84
Borsuk’s conjecture....Pages 85-90
Front Matter....Pages 91-91
Sets, functions, and the continuum hypothesis....Pages 93-108
In praise of inequalities....Pages 109-115
A theorem of Pólya on polynomials....Pages 117-122
On a lemma of Littlewood and Offord....Pages 123-126
Cotangent and the Herglotz trick....Pages 127-132
Front Matter....Pages 91-91
Buffon’s needle problem....Pages 133-136
Front Matter....Pages 137-137
Pigeon-hole and double counting....Pages 139-149
Three famous theorems on finite sets....Pages 151-155
Shuffling cards....Pages 157-166
Lattice paths and determinants....Pages 167-172
Cayley’s formula for the number of trees....Pages 173-178
Completing Latin squares....Pages 179-184
The Dinitz problem....Pages 185-190
Identities versus bijections....Pages 191-196
Front Matter....Pages 197-197
Five-coloring plane graphs....Pages 199-202
How to guard a museum....Pages 203-206
Turán’s graph theorem....Pages 207-211
Communicating without errors....Pages 213-222
Of friends and politicians....Pages 223-225
Probability makes counting (sometimes) easy....Pages 227-235
Back Matter....Pages 236-239