I'm planning to play with this problem together with some students today, and with my math salon tomorrow. Before starting to think about that, I was doing my usual early morning wandering through the blogosphere, and most amazingly landed on a post about this very same problem at 360. I'll get to 'borrow' the work Xi did putting together that post.
I'll also be 'borrowing' some great ideas I heard over 20 years ago in a math talk given by John H. Hodges. (I think it was at the NCTM conference in '81 or '82, and I think he was at UC Denver.) He used the Bridges problem to illustrate the steps he laid out in his 5 Major Steps in Mathematical Creation. I'd love to contact him to get his permission to post his notes (I have them on ditto!) here, but so far I haven't managed to find him.
Anyone have any favorite materials on this you'd like to point me to? I wouldn't mind borrowing a bit more... ;>