The empire problem, also known as the 
-pire problem) asks for the maximum number of colors needed
to color countries such that no two countries sharing a common border have the same
color (this is the usual four-color theorem)
in the case where each country consists of 
disjoint regions. Heawood (1890) showed that 
colors are sufficient, and for the case 
, 12 colors are also necessary
(Gardner 1997; Frederickson 2002, pp. 31-32).
Empire Problem
See also
Earth-Moon Problem, Four-Color TheoremExplore with Wolfram|Alpha
References
Frederickson, G. N. Hinged Dissections: Swinging & Twisting. New York: Cambridge University Press, 2002.Gardner, M. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New York: Springer-Verlag, 1997.Heawood, P. J. "Map Colour Theorems." Quart. J. Pure Appl. Math. 24, 332-338, 1890.Referenced on Wolfram|Alpha
Empire ProblemCite this as:
Weisstein, Eric W. "Empire Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EmpireProblem.html