Impossible Braids!
Here is a lovely mystery about braids:
Must one have three loose ends in order to make a braid?
To make sense of this question, look at this video: Braid Mystery.
For a resolution to the mystery see Braid Resolution.
For more on braids and teacup twists and other braid wonders see the June 2013 COOL MATH ESSAY. More still appears in chapter 3 of Math Galore!