One of the great things about the Möbius strip is that you can do simple things to it and get very unexpected results.
Make yourself a few Möbius strips out of paper. I usually chop up a piece of A4 paper into long strips about 4cm wide, joining the ends of each strip with a half-twist. Now get yourself a pair of scissors and let the fun begin!
1) First cut along the centre of the Möbius strip. What do you think will happen? Will there be two strips after the cut, or one? Will they be orientable or not? 2) Now do the same thing on another Möbius strip, but make your cut 1/3 of the way along the width of the strip. Is the result the same as before? If not, why not?
3) Make a Möbius strip where the ends are joined together with three half twists instead of one. Repeat experiment (1), cutting down the middle of the strip. What do you get?
I will post up the results and explanations in a few days’ time. Until then, I look forward to your comments and bafflements!