I want to make a presentation of Burnside's lemma outside of group theory, and more as the stand-alone combinatorial tool that it can also be. My plan right now is to make it into a 15-20 minute video, a la 3blue1brown, or Numberphile, or any of a number of other great educational math videos out there. I have no idea whether I will ever get that far, but that's the plan anyways.
My target audience would be more or less the same as those. Say "math interested high schoolers" to be concrete.
The plan is to establish an example set with some group action, and then derive Burnside's lemma for that particular example. I think I have the theoretical parts more or less figured out, but I'm missing a hook. A great example. Something "regular people" can actually encounter and want to reflect upon outside of a math puzzle.
I think it would be best to have an example using the dihedral group with 8 elements, or the full symmetry group on 3 elements. Either of those two groups would be a decent mix of non-trivial and still small enough to actually be able to work with.
I have thought about making necklaces / bracelets to appeal to the arts-and-craftsy people out there. I even got hold of a couple of A and B tags to actually make the necklaces. But it feels a bit too artificial, especially any restrictions on the number of tags to use (why exactly 3 letters, or exactly 4, when I clearly have at least 4 of each letter readily available, for instance).
I've thought about arranging people around a table for a game of Uno, where it matters who you sit next to, but not so much who's on the right and who's on the left, as play swaps direction all the time. But then the orbits on the set of possible table seatings are all max size anyways because no two people are the same.
I could to an entirely abstract example, with actual squares or triangles, with painting or marking the corners or sides, but I don't think that would be suitable as a motivational example.
Maybe I'm being too picky, and should just go with the bracelets. But I thought I should at least hear if anyone here had a great idea before I lock myself onto that track.