Number Theory 1. Six proofs of the infinity of primes …… 3. Binomial coefficients are (almost) never powers 4. Representing numbers as sums of two squares …… 8. Some irrational numbers 9. Three times π2/6 Geometry 10. Hilbert’s third problem: decomposing polyhedra63 11. Lines in the plane and decompositions of graphs …… 15. The Borromean rings don’t exist …… 17. Every large point set has an obtuse angle 18. Borsuk’s conjecture Analysis …… 21. The fundamental theorem of algebra …… 25. Cotangent and the Herglotz trick 26. Buffon’s needle problem Combinatorics 27. Pigeon-hole and double counting …… 32. Cayley’s formula for the number of trees 33. Identities versus bijections …… 36. The Dinitz problem247 37. Permanents and the power of entropy253 …… 41. Communicating without errors 42. The chromatic number of Kneser graphs …… 44. Probability makes counting (sometimes) easy
