This question came from trying to design some 3d printed dnd dice. I think it's possible to make a die unfair without changing the center of mass or affecting the exterior shape*, but I'm not sure exactly how to do this, or what the result would be. That being said, I have some theories:

• A cubic die would be the most influenced by one of two strategies:

1. A heavy rod connecting the centers of two opposite faces.
2. A disk (or ring) equidistant between two faces.

• If the internal weights look the same from when viewed from any side (after rotation/mirroring in the viewing plane), the die is fair. Some examples:
• A sphere at the center
• Weights with xy, yz and xz symmetry
• A rod connecting opposite corners, such as (-1,-1,-1) and (1,1,1)

So, is it possible?

*As a side note, only the vertices are necessary for a convex polyhedron to correctly collide with a flat surface. Some really cool 3d printed dice consist only of vertices connected to the center.