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    3-Regular Polyhedron: How many square faces can the polyhedron have?

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    5. The faces of a 3-regular polyhedron are all squares or hexagons. How many square faces can the polyhedron have? Does the number of square faces uniquely determine the polyhedron? (Please see attachment)

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    First of all, we have to state precisely that we are talking about convex polyhedrons.
    For every convex polyhedron (either regular or not) it is valid the Euler relation:

    v + f - e = 2 ( 1)
    v = no. of vertices
    f = no. of faces
    e = no. of edges

    Let denote ...

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    A 3-regular polyhedron is investigate. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.