Cases of the Four Color Map Theorem I've been searching for a specific case of the four color map theorem all over the internet, but can't seem to find a reliable explanation. 

Does anyone know one or two "cases" of maps as applied to the four color map theorem, including how it could be proved by hand? 

The only explanation given on the web is Appel &amp; Haken's computer-based proof, which is not helpful in the sense that the code used in their programs is not available to the public, nor do I understand it completely.

Also, can someone please explain (in depth) what Kempe's chains are and how they work? (maybe also provide an example?)

I am writing a paper for my math class about the four color map theorem, and so far the internet has not been very promising.
Kempe chains are explained in textbooks on graph theory. They appear in the five color theorem, an easy result which is proved using them.

The four color theorem is unfortunately rather complicated. You can try reading a summary of the Robertson et al. proof (including pointers to the computer code they use), Steinberger's simplified proof or Gonthier's account of his group's computer-checked proof.