As a grandfather, I always love being able to take time over the holidays to share entertaining math enrichments with the grandkids. Last Christmas I showed them this one and it was a big hit
This, to me, is the greatest Mobius related activity I have ever seen. I wrote about it briefly as part of a longer blog, but wanted to focus one on just this neat activity. (I have also included a link at the bottom to a nice Matt Parker video with a view others. I got this from an Ivars Peterson article in the New York Times.
Don't read the article until after you have tried it, He offers teasers of what the outcome could be.
Start by cutting out a cross of paper. Make the sides wide enough to do some cutting, one of them into thirds (sort of).
Now take ends of one cross and give them the standard half-twist to make a Mobius strip with a cross piece hanging on. Now take the other crossing pair and fold them away from the Mobius loop to make a regular (non-twisted) loop. It should look sort of like a twisted figure eight.
Now draw a line trisecting the Mobius branch. One line 1/3 of the way across the page should loop back around the loop and eventually make something like three paths on both sides of the strip.
Now cut along the trisected loop, then bisect the non-Mobius loop.
Shake out all the twists and turns to be amazed.
A while after I wrote this, I came upon a video of a talk by Matt Parker in which he includes several demonstrations kids (of all ages) would enjoy using Mobius strips of different numbers of twists, including zero twists.
So if you are looking for a way to share your love of math over the holidays with young people, you could work up a nice routine with some of these.