Two non-adjacent faces of a polyhedron are called $\textit{buddies}$ if they lie on the same plane. Call a polyhedron $\textit{nice}$ if every face has a buddy. What is the smallest $\textit{nice}$ polyhedron?
Any ideas?
I tried the case for a $\textit{nice}$ polygon and got the smallest polygon as the star (10 sides). This serves as a lower bound for the smallest $\textit{nice}$ polyhedron, as any cross-section of this should give a $\textit{nice}$ polygon.
I make no claims to minimality, but here's a nice polycube with $20$ cubes and $72$ faces:
We start with a $2\times2\times2$ green cube, then add two diagonally opposite red cubes on each face such that the diagonals on which the red cubes lie form the edges of an inscribed regular tetrahedron on the cube.
Amusingly, it bears a strong resemblance to the MSE logo.
(Though it's a less elegant solution, one can delete two of the externally-visible green cubes from this structure to reduce the number of faces to $60$ and things still work out. Removing all green cubes disconnects the polyhedron, so we need to leave some in.)
A computer search confirmed that no polycubes with $10$ or fewer cells are nice.