The four-color map theorem states that, at most, four colors are needed to color any map, if two adjoining countries are always to be assigned different colors. The statement of the theorem is deceptively simple: it says that every imaginable map--regardless of how complicated the countries are and how many countries adjoin any given country--can be colored using only four colors. For the purposes of this problem, two countries are considered to adjoin each other if the boundary that they share has non-zero length, so that two countries that only meet each other in a corner can be given the same color (this fits in with commonsense principles of map-coloring, since there is no danger of confusion when two countries that only meet at a corner have the same color). So for example, a chessboard can be colored using only two colors, black and...