The Five Color Theorem

Doodling on a map of England in 1852, Francis Guthrie noticed that only
four colors were needed to color the counties.  He conjectured that any
map could be colored with only four colors.   Several mathematicians
tried and failed to prove this; notably in 1879 Kempe published a proof
and only in 1890 was the flaw found by Heawood.  This four color conjecture
evaded a proof until 1972, when Appel and Haken gave a proof that required
a computer.  While there is as yet no Human readable proof, Kempe's
argument suffices to prove that five color suffice, and this gives a flavor
of known proofs of the four color theorem.  I will sketch this history
and prove the five color theorem, and then discuss the coloring theorem
for other surfaces (torus, projective plane, Klein bottle...)