Finding a Hamiltonian Path in a Cube with Specified Turns is Hard

We prove the NP-completeness of finding a Hamiltonian path in an N ×N ×N cube graph with turns exactly at specified lengths along the path. This result establishes NP-completeness of Snake Cube puzzles: folding a chain of N unit cubes, joined at face centers (usually by a cord passing through all the cubes), into an N ×N ×N cube. Along the way, we prove a universality result that zig-zag chains (which must turn every unit) can fold into any polycube after 4×4×4 refinement, or into any Hamiltonian polycube after 2× 2× 2 refinement.