Treebeard's Stumper Answer

Tangled Up Puppet
We're making string puppets in my afternoon Tools & Materials class at Dunn Middle School! I was inspired by the gift of a hopelessly tangled puppet from Mexico. I've noticed that my puppetinprogress also has an uncanny knack for getting tangled up, and that's the stumper. Consider the simplified threestring puppet shown below. It's easy to make your own with soft cord like shoelaces and a piece of cardboard. Starting with straight untangled strings, I can make each of these braids with just two moves of the puppet piece. Can you tangle and untangle them?
Can you tangle and untangle these puppet strings? You should assume the
top piece is fixed and can't be manipulated, but you can do anything with
the bottom puppet except untie it. How many different moves are possible?
I figure every tangle is just a succession of simple moves one after the other.
To explore this stumper, you should make a simple threestring puppet like mine. I cut the plaque from a scrap of thin plywood, but cardboard works. Be sure you can tell which side is which.
The tangle on the right is especially interesting since it is a classic hair braid. Yes, it's possible to make this with a puppet that's fixed above and below. That means it's also possible to braid your hair without untying it! It's also possible to braid a solid piece of leather, though it looks like it was done in hyperspace! Note that the braided strands are all sameside up.
The hardest part of this stumper might be describing the moves in words. Creating graphics for the moves won't be much easier. But there are a limited number of moves with a threestring rigid puppet, so it should be possible to describe them all. Puppets get tangledup with one simple move after another, and it must be possible to undo any tangle in the same way!
My fellow Dunn School teacher Dorothy gave me this charming tangled up (and blue) puppet from Mexico a few years ago. I was ready to cut the strings and reassemble it, but Julie managed to untangle it without any cutting. In principle, that's not surprising. If the strings got tangled without cutting, then it must be possible to untangle them with simple moves in the right order. My "Treebeard" puppetinprogress is on the right, facing a twoweek deadline.
"Tangled Up Puppet" is a classic song by Harry Chapin. What other puppet songs can you find?
There's an odd kind of algebra in tangled up puppets. I believe there are just six basic ways to rotate the threestring puppet. You can turn it sideways to the right or left (call it RR and RL). You can also rotate it forward or backward up between strands 1 and 2 (F12 and B12) or strands 2 and 3 (F23 and B23). The middle tangle in last week's stumper can be made by the simple operations {F23 + B12}, and it can be undone with {F12 + B23}. Braid Theory is an interesting math without numbers.
Notes:
I should know better than to ask such a hard stumper at the busiest time of the school year, but I figure posing a good question is as important as answering it. My DMS student Katie is a master at untangling braids and puppets, but how does she do it? I never did get back to this stumper after the endoftheyear crunch at school, and now it's a tangled stumper for me too! I've got new stumpers on my mind, but I can point you in the right direction. Have fun!
 I started thinking about this stumper with two classic articles by Martin Gardner: "The Church of the Fourth Dimension" in The Unexpected Hanging (1969) and "Group Theory and Braids" in New Mathematical Diversions (1971).
 Colin Adams' The Knot Book (2001) is "An Elementary Introduction to Mathematical Theory of Knots", but it's not easy!
 There are many links on Knot and Braid Theory if you do a web search, and even some free software. Here are some starting links:
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