Consider the cubic 3-dimensional torus T^3 = R^3/Z^3, a unit cube with opposite faces identified. Now connect each of 4 alternate corners of the cube, say {(1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1)}, to the center (1/2,1/2,1/2) of the cube by a line segment. Now we want to shorten this union of 4 segments with these moves: 1) Any vertex (endpoint of a segment) may be moved continuously, while keeping the same connectivity (segments that connect 2 vertices will still connect them even if they are moved); and 2) Any vertex v may split in two to become 2 new vertices v', v''. v' and v'' are assumed to be connected by a line segment of length 0. When that happens, one or more segments that had v as an endpoint (but not all of them) are assigned to have v' as endpoint, and the remaining segments are assigned to have v'' as endpoint. 3) The reverse of 2): any 2 vertices may (by 1) coalesce to form a new vertex. Puzzle: Using only these moves, what is the shortest possible network that the original network of 4 segments can be transformed to? —Dan