A set of 12 rods, each 1 metre long, is arranged so that the rods form the edges of a cube. Two corners, $A$ and $B$, are picked with $A B$ the diagonal of a face of the cube.
An ant starts at $A$ and walks along the rods from one corner to the next, never changing direction while on any rod. The ant's goal is to reach corner $B$. A path is any route taken by the ant in travelling from $A$ to $B$.
(a) What is the length of the shortest path, and how many such shortest paths are there?
(b) What are the possible lengths of paths, starting at $A$ and finishing at $B$, for which the ant does not visit any vertex more than once (including $A$ and $B$ )?
(c) How many different possible paths of greatest length are there in (b)?
(d) Can the ant travel from $A$ to $B$ by passing through every vertex exactly twice before arriving at $B$ without revisting $A$ ? Give brief reasons for your answer.