Question on convex polyhedra - how would I approach this?

Question

Question:

For which n is there a convex polyhedron with n regular (possibly different) faces?

How would one approach such a question?

(This question was one that I was given a few weeks ago as part of a set of other questions to solve for fun)

So far on this topic, I’ve learnt about Euler’s formula and a little on diagonals, but not much else.

Answer 1

The $n$-gonal prism has $n+2$ faces, so that settles the problem for $n\ge5$. The tetrahedron has $4$ faces and it is trivial to show that no polyhedron can have fewer faces. So the polyhedra you want exist for all numbers of faces where any kind of polyhedron exists, i.e. at least $4$ faces.