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 27-36
Front Matter....Pages 37-37
Hilbert’s third problem: decomposing polyhedra....Pages 39-45
Lines in the plane and decompositions of graphs....Pages 47-51
The slope problem....Pages 53-57
Three applications of Euler’s formula....Pages 59-64
Cauchy’s rigidity theorem....Pages 65-68
Touching simplices....Pages 69-72
Every large point set has an obtuse angle....Pages 73-78
Borsuk’s conjecture....Pages 79-84
Front Matter....Pages 85-85
Sets, functions, and the continuum hypothesis....Pages 87-98
In praise of inequalities....Pages 99-105
A theorem of Pólya on polynomials....Pages 107-114
On a lemma of Littlewood and Offord....Pages 115-118
Cotangent and the Herglotz trick....Pages 119-124
Buffon’s needle problem....Pages 125-128
Front Matter....Pages 129-129
Pigeon-hole and double counting....Pages 131-141
Three famous theorems on finite sets....Pages 143-147
Lattice paths and determinants....Pages 149-154
Cayley’s formula for the number of trees....Pages 155-160
Completing Latin squares....Pages 161-166
The Dinitz problem....Pages 167-172
Front Matter....Pages 173-173
Five-coloring plane graphs....Pages 175-178
How to guard a museum....Pages 179-182
Turán’s graph theorem....Pages 183-187
Communicating without errors....Pages 189-198
Of friends and politicians....Pages 199-201
Probability makes counting (sometimes) easy....Pages 203-211
Back Matter....Pages 212-215