Mobius strips are easy to make. Just glue one end of a strip of paper to the back side of the other end. Then you have a twisted strip.

Lots of knitters make Mobius cowls as well. It’s a popular and cozy style.

The author of some of the books I’ve used with students, mathematician Richard Evan Schwartz at Brown University, has been puzzling over a problem posed in the 1970s: what’s the shortest strip of paper you can use to make a Mobius strip?

After a few years of tinkering, Dr. Schwartz cracked the problem. The key to doing so: taking a paper mobius strip and cutting it, finding that the resulting shape was not a parallelogram as he thought, but rather a trapezoid. Then he was able to calculate what he needed to complete the proof. Scientific American wrote a story about this, where you can read more.

This is yet another example of the ways hands-on experience can lead to breakthroughs even in advanced math!

Dr. Schwartz also allowed me to reference his wonderful book You Can Count on Monsters in my upcoming book, with the working title The Art of Learning Math: A Handbook for Success. Read my review of his book here.

Update: Dr. Schwartz provided an explanation of his proof, with illustrations and activities, that high school students should be able to work with. You can access this here (free).