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MatheMUSEments
Multicolored Maps
By Ivars Peterson
Muse, February 2002, p. 44.
Maps of the United States often show the states in different
colors. In general, mapmakers use enough colors to make sure
states that touch are never the same color.
Suppose you have only three colors of pens. Is that enough to
fill in all the states? Not quite. Nevada, for example, is
surrounded by five other states: California, Oregon, Idaho, Utah,
and Arizona. There's no way to color this group of states without
using a fourth color. Can you figure out what other states force
the use of a fourth color?
Would you ever need five colors? In 1976, two mathematics
professors at the University of Illinois proved that four colors
would be enough for any map that could be drawn on a flat piece
of paper.
Curiously, four colors are needed whenever a state is
surrounded by three or more states and the number of surrounding
states is an odd number. You see that not only in a U.S. map but
also in a map of South America. The country Bolivia, for example,
is surrounded by an odd number of countries: Peru, Brazil,
Paraguay, Argentina, and Chile. What other country in South
America is in the same situation?
Simpler maps sometimes use fewer than four colors. For
example, if you had a map that looked like a checkerboard or some
other sort of grid, you would need only two colors.
What about a pattern made up of tiles shaped like kites and
darts or a pattern made of diamonds like the one below?
Mathematicians have recently proved that you need at most
three colors for such a map.
Answer to U.S. map: Kentucky, which is surrounded by seven
states; and West Virginia, which is surrounded by five states.
Answer to South American map: Paraguay, which is surrounded by
three countries.
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