Four non-overlapping spherical planets of identical radius are situated in space. The surface of each planet is colored either white or black. The surfaces are white wherever that point is visible to an observer on another one of the planets. They are black where no other planet can see that part of the surface.
What is the total area of the black surfaces? What if there were 10 planets instead of 4?
If the answer does not depend on the actual configuration, as one must assume it does not, the answer must be equal to the surface area of one of the planets. This can most easily be seen in the case where the planets are arranged in a rectangle, where one quarter of each of the four planets is black; or in the case where all four planets are in a straight line, where 1/2 of each of the two end planets is black.
Similar considerations apply in the case of 10 planets: arranged in a regular decagon, 1/10 of each of the 10 is black; arranged in a straight line two half-planets are black. In either case, one whole planet's worth of surface is black.
Posted by Charlie
on 2013-08-08 16:09:25