Solution: Start at the 3-1 end of the string (on the left side of the photo). Place the first branch in the right upper front corner of the 3x3x3 cube. Denote the direction from left to right by +0, from left to right by -0, front to back by +2, back to front by -2, from down to up by +1, and up to down by -1. Then take the directions

   -1  -0  +1  -2  -0  +2  -1  -2  +0  +1  -0  +2  -0  -2  +0  -1  -0