If you've constructed a Moebius strip (it's not too hard), you can have even more fun (tm) destroying it! It's amusing to try to guess what these two cuts come out as, so I won't say.
- Take a Moebius strip and carefully cut along the strip's centre line. The cut goes once around, and you end up with the scissors reversed relative to the strip. But that's not the fun part!
- Take another Moebius strip (the old one was ruined, if you look carefully), and cut it along the one-thirds line. This time the cut goes twice round the strip before it meets up with itself. Don't drop anything!