Cut out a strip of paper, say two inches by ten inches. Color the two sides of the strip with different colors. I am going to use yellow for the "front" and red for the "back". Mark a point on the "front," and a second point on the "back". Obviously, there is no way you can join these two points with a continuous line.

Now twist the strip once.

And glue the ends together.

Notice how, at the line of gluing, red meets yellow. Trace your finger around the strip. Convince yourself it has no back and no front, but only one side. Can you see that you can now join your two different points with a continuous line?

Further food for thought:

• Cut your band along its middle. You should get one loop with two sides.
• If you cut this new loop down its middle you should get two linked loops, each with two sides.
• In Tiling the Plane we met the Four-Color problem. How many colors would you need to use for a map drawn on the surface of a Möbius band?
• The University of Wales at Bangot has an excellent site which includes more cool information on möbius bands.