8. mapmakers have long believed that only four colors are necessary to distinguish among any number of different
countries on a plane map. countries that meet only at a point may have the same color provided they do not have
an actual border. the conjecture that four colors are sufficient for every conceivable plane map eventually attracted
the attention of mathematicians and became known as the "four-color problem". despite extraordinary efforts over
many years to solve the problem, no definite answer was obtained until the 1980's. four colors are indeed
sufficient, and the proof was accomplished by making ingenious use of computers. the following problems will
you appreciate some of the complexities of the four color problem. for these "maps", assume that each closed
region is a different country. what is the minimum number of colors necessary for each map?