In every polyhedron there is at least one pair of faces with the same number of sides.
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Copyright © 19962018 Alexander Bogomolny
In every polyhedron there is at least one pair of faces with the same number of sides.
Let N be the greatest number of sides in a face of a given polyhedron. Then the are N adjacent faces each having the number of sides between 3 and N. There bound to be 2 with the same number of sides.
Please note that we have actually proved a stronger statement.
References
 I. F. Sharygin, Mathematical Mosaic, Mir, 2002, problem 65.4 (in Russian)
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Copyright © 19962018 Alexander Bogomolny