I had the desire to create a cube out of a single piece of string, where each edge is represented only once. Through experimentation it appears that this is impossible, and the closest you can get is to create a cube missing 3 of the 12 edges.
A tetrahedron leaves 1 left over. An octahedron leaves none. Why is the limit 3 for a cube, and 1 and 0 for those? Is there a general rule that predicts this? What about an icosahedron, or a tesseract?