In every polyhedron there is at least one pair of faces with the same number of sides.

Solution


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Copyright © 1996-2018 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

  1. I. F. Sharygin, Mathematical Mosaic, Mir, 2002, problem 65.4 (in Russian)

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Copyright © 1996-2018 Alexander Bogomolny
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