The thing that I found most interesting was actually the little side discussion about all mazes being two pieces. Something that I had never considered, but seems fairly self evident (assuming only one path exists through the maze). Also makes me wonder what a proof would look like.
> Also makes me wonder what a proof would look like. Start with the simplest possible maze (i.e. one square with a start and a finish) which will be in two pieces. Then show that all possible additions will not add further pieces except for when you "cut" into an existing piece to make a separate non-solution route (i.e. a looped route that takes you back to the same point). (Having any loops in the maze would break…
The loop could be somewhere inaccessible from the starting point. Perhaps your logic holds if you insist that all pathways in the maze are connected, though.