Ref:
Definitions: A polygon P has an inside-out dissection into another polygon P' if P′ is congruent to P, and the perimeter of P becomes interior to P′, and so the perimeter of P′ is composed of internal cuts of the dissection of P. A dissection from P to P' is totally (or fully) inside out if we further insist that no point on the boundary of P should be on the boundary of P'
Question: A cube can obviously be dissected via 8 smaller cubes into an inside-out cube. What is the least number of intermediate polyhedral pieces if we need to dissect a cube totally inside-out into another cube (that such a dissection is possible is easy to see: eg: one could cut the cube into a large number, say 8000, of small identical cubes and exchange all 'surface cubes' with 'core' ones)?
Note: Same question can be asked reg inside-out / totally inside out dissections of a tetrahedron (regular or arbitrary) into a tetrahedron congruent to itself - I don't know if every tet can be inside-out / totally inside-out dissected into a tet congruent to itself via convex intermediate pieces. Second reference above asks about dissecting a tet into some convex polyhedron.