Build a polyhedron by taking lots of 1X1X1 cubes and gluing them together face-to-face in some interesting way. Can you unfold this polyhedron by cutting along edges to get a single flat shape with no overlaps?

More specifically, a polycube is a polyhedron built out of lots of identical cubes glued together face-to-face in any way you choose. Having built such a polyhedron, now forget about the interior faces, the faces where two cubes get glued together, and just think of the polycube as built out of the square faces you see on the outside. There is one more requirement we need to impose, which is that the polyhedron should be deformable to a sphere. In practical terms, this means that if you made it out of rubber (with the interior faces removed) and pumped it full of air it would inflate to a ball, not an inner tube or a pretzel shape, for example. In other words, there shouldn't be any holes going right through the polyhedron. See the illiustrations for examples.

The question asks whether every polycube has an edge-unfolding: An edge-unfolding is what you get you cut the surface of the polyhedron up along some (but not all) of the edges of the various square faces, and then unfold what you get into one big piece that lies flat on the table. Can you do this so that the unfolded pattern doesn't overlap itself on the flat table?

Another way to say this is: Suppose you've built the polyhedron out of lots of cubes glued together, and now you want to wrap it up tightly in paper. Can you cut out one single shape out of a piece of paper (this shape will be made out of lots of identical squares) so that you can wrap up the polyhedron by just folding up this shape around the polyhedron face by face?