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The magician tries to fit a loop of paper around his wrist (or around the magic puppet's neck) but it won't fit. The magician says, "Hmmm, I'll have to cut this loop bigger". The magician takes a pair of scissors and cuts the loop in half up the middle. Instead of two loops, the magician ends up with one larger loop that now fits around his wrist!
(Normally, you would expect a loop cut in half up the middle to turn into two loops, instead of one big loop).
OPTIONAL: Have an audience member cut a regular loop (one without a twist) in half up the center before you start your trick -- they will get two loops. Before cutting your loop, you can ask the audience to guess what you will get if you cut the loop in half up the center... As long as you don't have anyone in the audience who knows about Mobius Strips you should get lots of guesses like "you'll end up with two loops".
Take a long, fairly wide strip of paper. Twist the paper once and tape it into a loop.
This type of loop is called a "Mobius Strip".
Snip the loop up the center.
Don't go too fast (snip, snip, snip) and keep as close to the exact center as you can.
You can babble some "magic words" as you go (Hocus Pocus, Luminous Mobius are the ones I like!)
When you're done cutting, you'll end up with one big loop with a couple of twists in it -- the natural assumption is that you'd end up with two loops.
A loop with a single twist in it is called a mobius strip. The "Mobius Strip" is an actual mathematical phenomenon. You aren't really doing magic, you're doing math!
The Möbius strip has several curious properties. A line drawn starting from the seam down the middle will meet back at the seam but at the "other side". If continued the line will meet the starting point and will be double the length of the original strip. This single continuous curve demonstrates that the Möbius strip has only one boundary. The example they always give in university is that if an ant walks along the edge of the strip, he'll travel twice as long as the loop before he gets back to his starting point.
Cutting a Möbius strip along the center line yields one long strip with two full twists in it, rather than two separate strips. This happens because the original strip only has one edge which is twice as long as the original strip.
If you're confused right now, don't worry -- I am too. Math was never my strong suit -- suffice it to say just like gravity, mobius magic works whether you understand it or not!
When you do the trick, you have to be careful to cut as close to the center as you can, because there's a second magical mathematical ability the Mobius Strip has. If the strip is cut about a third of the way in from the edge, it creates two strips: One is a thinner Möbius strip, the other is a longer but thin strip with two full twists in it. So keep your cut close to the center so you don't accidentally end up with this.
Grandma asked me... Yes, but now how do we explain this all to a five year old!? The answer is, you don't have to... You can just tell them it's magic!