A number of stationary perfectly-spherical planets of equal radius are floating in space. The surface of each planet includes a region that is invisible from the other planets. Show that the sum of these regions is equal to the surface area of one planet.
Fix any direction and call it “north.” Look at the north poles of all planets. A north pole is private if and only if there are no planets further to the north. Therefore, only the northernmost planet has a private north pole.
Since north was arbitrary, this is true for any direction. If we take the surface a single planet, every point on that planet corresponds to a direction. Thus each point on this surface maps to exactly one hidden point - the hidden point for that direction. Similarly, all hidden points map to a unique direction and thus a unique point on the surface of the planet.