Regarding the "four color problem":

The problem is not that MORE than four colors are needed. It is just the opposite. Why are more than four colors NEVER needed? The convention among mapmakers is that areas on a map that share NON-POINT boundaries must be given different colors. For example, a map of Great Britain would need two colors. (Wales and Scotland would be the same color because they do not share a boundary.) Now consider a map of Belgium and her neighbors. You need four colors: one for Belgium, another for France, a third for Germany, and a fourth for Luxemburg. You could use the same color for Luxemburg and the Netherlands, since they have no common boundary. Can you design a map that needs FIVE colors? Sound easy? Try it sometime! It has never been done. Even if you do not limit yourself to real geography, you probably can't do it. No one ever has. The "four-color problem" is the attempt to prove mathematically that four colors are all that are EVER needed. Please note that, for purposes of examining the four-color problem, discontinuous areas are considered separately. For example, in a North American map, Alaska and Lower 48 would be treated as if they were "different countries". There is apparently some controversy as to whether or not the problem has been solved mathematically.
Kent