Proofs from the book

Number theory: Six proofs of the infinity of primes ; Bertrand's postulate ; Binomial coefficients are (almost) never powers ; Representing numbers as sums of two squares ; The law of quadratic reciprocity ; Every finite division ring is a field ; The spectral theorem and Hadamard's determinant problem ; Some irrational numbers ; Three times [pi squared]/6

Geometry: Hilbert's third problem : decomposing polyhedra ; Lines in the plane and decompositions of graphs ; The slope problem ; Three applications of Euler's formula ; Cauchy's rigidity theorem ; The Borromean rings don't exist ; Touching simplices ; Every large point set has an obtuse angle ; Borsuk's conjecture

Analysis: Sets, functions, and the continuum hypothesis ; In praise of inequalities ; The fundamental theorem of algebra ; One square and an odd number of triangles ; A theorem of Pólya on polynomials ; On a lemma of Littlewood and Offord ; Cotangent and the Herglotz trick ; Buffon's needle problem

Combinatorics: Pigeon-hole and double counting ; Tiling rectangles ; Three famous theorems on finite sets ; Shuffling cards ; Lattice paths and determinants ; Cayley's formula for the number of trees ; Identities versus bijections ; The finite Kakeya problem ; Completing Latin squares

Graph theory: The Dinitz problem ; Permanents and the power of entropy ; Five-coloring plane graphs ; How to guard a museum ; Turán's graph theorem ; Communicating without errors ; The chromatic number of Kneser graphs ; Of friends and politicians ; Probability makes counting (sometimes) easy.

Number theory. Six proofs of the infinity of primes

Bertrand's postulate

Binomial coefficients are (almost) never powers

Representing numbers as sums of two squares

The law of quadratic reciprocity

Every finite division ring is a field

Some irrational numbers

Three times [pi squared]/6

Geometry. Hubert's third problem: decomposing polyhedra

Lines in the plane and decompositions of graphs

The slope problem

Three applications of Euler's formula

Cauchy's rigidity theorem

Touching simplices

Every large point set has an obtuse angle

Borsuk's conjecture

Analysis. Sets, functions, and the continuum hypothesis

In praise of inequalities

The fundamental theorem of algebra

One square and an odd number of triangles

A theorem of Pólya on polynomials

On a lemma of Littlewood and Offord

Cotangent and the Herglotz trick

Buffon's needle problem

Combinatorics. Pigeon-hole and double counting

Tiling rectangles

Three famous theorems on finite sets

Shuffling cards

Lattice paths and determinants

Cayley's formula for the number of trees

Identities versus bijections

Completing Latin squares

Graph theory. The Dinitz problem

Five-coloring plane graphs

How to guard a museum

Turán's graph theorem

Communicating without errors

The chromatic number of Kneser graphs

Of friends and politicians

Probability makes counting (sometimes) easy.

Number Theory

1. Six proofs of the infinity of primes

2. Bertrand's postulate

3. Binomial coefficients are (almost) never powers

4. Representing numbers as sums of two squares

5. Every finite division ring is a field

6. Some irrational numbers

7. Three times [pi][superscript 2]/6

Geometry

8. Hilbert's third problem: decomposing polyhedra

9. Lines in the plane and decompositions of graphs

10. slope problem

11. Three applications of Euler's formula

12. Cauchy's rigidity theorem

13. Touching simplices

14. Every large point set has an obtuse angle

15. Borsuk's conjecture

Analysis

16. Sets, functions, and the continuum hypothesis

17. In praise of inequalities

18. theorem of Polya on polynomials

19. On a lemma of Littlewood and Offord

20. Cotangent and the Herglotz trick

21. Buffon's needle problem

Combinatorics

22. Pigeon-hole and double counting

23. Three famous theorems on finite sets

24. Shuffling cards

25. Lattice paths and determinants

26. Cayley's formula for the number of trees

27. Completing Latin squares

28. Dinitz problem

29. Identities versus bijections

Graph Theory

30. Five-coloring plane graphs

31. How to guard a museum

32. Turan's graph theorem

33. Communicating without errors

34. Of friends and politicians

35. Probability makes counting (sometimes) easy.