Suppose the sphere S^2 is made of (2D) square pieces of rubber sewn together abstractly along common edges. (I.e., the result is topologically equivalent to a sphere.) According to this rule, there is a trivial case where only 2 squares are sewn together, by identifying corresponding edges — this is indeed a sphere topologically. The fact that [in 3-space any two planar squares with the same boundary are the same] is not our concern. It's seems clear that — except for this trivial example — the smallest number of squares in such a thing is 6. Now suppose the 3-sphere S^3 = {x in R^4 | ||x|| = 1} is, similarly, built abstractly out of (solid) cubes, copies of Q = [0,1]^3, that are allowed to intersect only along entire common square faces, edges, or vertices. Puzzle: ------- What is the smallest number of cubes with which it's possible to build something topologically equivalent to S^3 this way? ------ I think I know, but I haven't tried to prove it yet. —Dan