One can form a polygon of $4 n$ sides by intersecting $n$ congruent squares (treated as closed sets, i.e., filled squares):
Q1. For which of the $k=3,4,\ldots,4n$ can the intersection of $n$ congruent squares result in a $k$-gon? Perhaps not all can be achieved?
Q2. Can the intersection of $n$ congruent cubes result in a polyhedron of $6 n$ faces? I believe so, but an explicit construction would be useful.
Q3. For which $k$ can $k$-face polyhedra result from the intersection of $n$ congruent cubes?
Q4. The questions extend to $\mathbb{R}^d$.
Question Q2 in particular occurred to me as a possible exercise to build 3D intuition.
Added. Here are two cubes intersected to produce a polyhedron of 12 faces (although one can hardly verify that from this single, not well-lighted image!):