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Paul Brown, an Australian math teacher and author of Proof, a book that I may characterize as a well-written guided introduction into that most fundamental activity, has brought to my attention a recent post at the Futility Closet blog, The Jeweler’s Observation, which I fully reproduce below:

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

The solution is initially hidden but becomes available on a button click:

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, Springer Verlag, 1999.

This example provides nice references to two powerful tools in the arsenal of mathematical problem solving: the Extremal and Pigeonhole Principles. However, that was not the reason I decided to write this blog. One of the most (if not the most) important strategy in problem solving, namely, G. Polya's Looking Back step.

So, looking back at the quote from A. Engel's book, what has been actually proved? From the proof we can extract a stronger assertion than that claimed in the problem. This can be formulated in a couple of equivalent ways. But first note that the convexity of the polyhedron was not absolutely essential and that condition can be weakened. For example, the argument appears to work for any polyhedron in which two faces may only share an entire edge (I am not sure even this is necessary). Thus, thinking of such polyhedrons, the original problem can be so generalized:

• Every polyhedron has a face with two adjacent faces that have the same number of edges.
• Every polyhedron has two faces with the same number of edges adjacent to the same face.

On a second look back, what is essential for the proof is the requirement that all faces of the polyhedron be convex polygons. If that condition is not satisfied, there are 3D objects with a "skeleton" formed by straight line segments, and "faces" bounded by these edges, with no two faces having the same number of edges. Can you give an example? (That would be a counterexample to the original claim.) I can think of a 3D object that consists of four pieces that have $3,4,5,$ and $6$- sided faces, the ones with $4$ and $5$ sides not being convex, or even flat.