Following up on my post with the subject line "(x^6-1)/(x^2-x+1) via pebbles", Joshua Zucker gave the URL of the video I was asking about:

http://www.mathlesstraveled.com/?p=81

Thanks, Joshua! So now that I have that link, I can ask my real question: What's the thoroughly modern way to work on puzzles like this? One can make physical models, but that's just so twentieth century. (Or maybe I should say "so nineteenth century": think of all those white plaster models of arcane surfaces that are sitting around in seldom-open display-cases in dozens of math departments around the country.) Is there any kind of virtual environment in which one can play with puzzles like this? I'm thinking of something like Harold and his purple crayon, but Harold could only draw static objects. I think I've heard about a "purple crayon physics" computer game in which one draws objects that come to life; that's more the sort of thing one would need. One advantage of a virtual environment is that it could keep track of what I've done, so I wouldn't be in the frustrating situation of having solved a puzzle but not remembering how I did it! (Which reminds me: I once managed to tie two mirror-image trefoil knots in a rope and pass one of them through the other in such a way as to make them cancel, leaving a completely unknotted rope. Now of course this is topologically impossible, as Conway showed in a very elegant fashion; so what I really came up with is some way of *appearing* to disprove Conway's theorem. This would make a fun little party trick for confusing drunk topologists. Trouble is, I can't remember how I did the trick!) Jim Propp