A compound of n cubes is a non-convex polyhedron formed from the union of n identical concentric cubes. There are many ways to create a compound of n cubes with overall polyhedral symmetry. We consider four particular cases of a compound of n cubes, where n ranges from 2 to 5, inclusive. These polyhedra are dissected into interlocking pieces, making assembly puzzles. We discuss the design of such puzzles.