r6's post over here got me reading Wikipedia a bit about the four-color theorem. Snarks and Planarity Criteria and Finite Minimal Obstruction Sets, oh my. Also there is an equivalent characterization of 4CT that I find remarkable, due to this guy. I may have read this in the past somewhere in John Baez's notes, but it's worth repeating:Theorem (Kauffmann, 1990): Observe that the cross product of vectors in R^3 is not associative. However, for any two parenthesizations of the product of variables A1, ... , An, for instance (A1 x A2) x (A3 x A4) and A1 x ((A2 x A3) x A4), there is some assignment of the positive unit vectors {i,j,k} to the variables A1...An so that the two products are equal and nonzero.
Only known proof: by reduction from 4CT
Also: if you can prove this theorem simply, it would give a simple proof of 4CT
Crazy!
