Why is the four coloring theorem so hard to prove when the five/six theorem proofs are much more accessible? I might be giving a talk to high school students soon. I plan to show them the proof for the six/five coloring theorems and also give a brief discussion of the famous four color theorem.
Why is the four color theorem so much harder to prove than the six/five color theorems? Is there a rather elementary reason for this?

The chromatic number of a graph imbedded in a genus p surface is $ \chi(S_p)  =  {\left\lfloor \frac{7 + \sqrt{1 + 48 p}}{2}\right\rfloor} , p \ge 1 $ where ${[x]}$ is is the largest integer not greater than $x$.     I was fortunate enough to take Gerhard Ringel 's  class  on this subject, ( he proved equality in the formula  in 1968) . I would still recommend "The Map Color Theorem" as a text in combinatorics . We did not study the proof of the case $ p = 0 $ of course where the chromatic number remarkably fits the formula but proof had been absent until Haken - Appel and the computer proof of 1976. I have no idea why things progressed in that manner, it's a very good question. The history of the proof is a little like the history of the Poincare conjecture. 
