November 2006: Afghan Bands

The famous one-sided Mobius strip has appeared in the artwork of M. C. Escher and is the basis for the recycling symbol. It was also the inspiration for an old magic trick called the "Afghan bands" which dates back to at least 1882. A circus magician holds up three loops of fabric, which he explains are cloth belts. The problem, he laments, is that he needs belts for two clowns, the fat lady, and the Siamese twins. He takes the first band, rips it down the centerline and produces the belts for the two clowns. He rips the second band in the same way, but instead of two loops, he holds a single loop that has twice the circumference of the original---the belt for the fat lady. Finally, to get the belts for the twins he rips the third loop and obtains two belts that are linked together. The trick, as we see below, is that the loops have twists in them (zero, one, and two half twists, respectively). For maximum effect, the fabric or paper should be flexible and be much narrower than it is long so that the audience does not notice the twisting.It turns out that one of the Siamese twins has put on a lot of weight over the summer. This month's problem is to find a way to repeat the Afghan bands trick, but at the end obtain one large belt linked with one smaller belt. Your solution should describe how to perform the magic trick.

If you liked this puzzle, try this one. Run a strip of paper through a wedding ring and tape the ends together to form a twisted band. Using only scissors (and no tape) cut the band so that the resulting band is knotted around the wedding ring (see below).

Dickinson College students can submit answers to Dave Richeson or Barry Tesman. The list of solvers will be posted at the end of the month.