The Jeweler’s Observation

Prove that every convex polyhedron has at least two faces with the same number of sides.

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Consider the face with the largest number of sides. If that face has m sides, then it’s surrounded by m faces. Any face must have at least 3 sides. So altogether in this group we have m + 1 faces, and each face must have between 3 and m sides. At least two faces must have the same number of sides. From Arthur Engel’s Problem-Solving Strategies, 2003.