Because the problem was preying on my mind last night, I had a nightmare. I was working on some other puzzle which had to do with billiard balls. So I somehow wound up in a pool game. Winning the game had something to do with figuring out the puzzle. Some other things happened that I don't remember very well, but one of the billiard balls got knocked off of the table. I didn't want to pick it up, because it looked funny (it was covered with dust), and I thought it was going to do something to me. But the other players insisted that I pick the ball up, so I did, and it bit me. Turns out there was a spider in the dust. That's where the dream ended. I woke up about 4:30am, and had some trouble falling asleep again.
The problem was still bothering me while I practiced piano. I couldn't concentrate very well and wound up making mistakes I hadn't made in a while. I definitely felt like I had lost some progress.
An upside to all of this is while googling I found some interesting links to sites that discuss Martin Gardner's books. (Some of them have solutions to problems that aren't in the books.) I thought I'd point out one site in particular from Jill Britton, who is a maths instructor at Camosun College in Victoria, British Columbia. Anyway, one of her links, Topology for Tots, discusses how it's bettet to teach topology before geometry to the very young. It reminded me of something that Herman Rubin, the statistics professor from Purdue complains about a lot in sci.math, misc.education, and some other places: that it's better to learn the general before the specific and that kids in the US are being ruined when this doesn't happen. I don't know enough about topology to have an informed opinion, but one thing that comes to mind is that a young child might very well have intuition about something but lack the language skills to express their understanding formally.