Planets


Consider a group of spherical equal (congruent) planets in space. Mark all the points on any planet which are not visible from any other planet. Find the sum of areas determined by these points.

Translate all the planets (at least in an abstract way 🙂 ) onto one single planet, and watch the mentioned marked points. The marked zones on two palnets cannot overlap in the translation, because if that happens, one of those points would be visible from another planet. Secondly note that all the points on the abstract planet are marked because looking from any spatial direction you can see one closest planet. Therefore the sum of the areas is equal to the area of one single planet.

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