A problem by Russian mathematician Viktor Prasolov: This hexagon has been divided into black and white triangles so that any two triangles either have a common side (in which case they’re painted different colors) or a common vertex or they have no points in common, and each side of the hexagon is a side of one of the black triangles. Prove that no such partition exists for a 10-gon.
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Well, suppose it were possible and that we’d done it. Let n be the number of sides of black triangles and m the number of sides of white triangles. Except for the 10 sides of the decagon, each side of a black triangle is also the side of a white triangle, so n – m = 10. But both n and m must be divisible by 3. That’s impossible, so the partition we’re seeking cannot exist.
(From Prasolov’s Problems in Plane and Solid Geometry.)
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