Ref: 
- https://mathoverflow.net/questions/186332/inside-out-polygonal-dissections

- https://mathoverflow.net/questions/443160/inside-out-dissections-of-solids

**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 can be seen by cutting the cube into a large number, say 8000, of small identical cubes and moving the 'surface cubes' inside)?

**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. Second reference above asks about dissecting a tet into some convex polyhedron.