A good example to show group actions and Burnside's lemma

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.

Well, certainly nearly all the usual texts (here is an open source one!) do this via necklace/bracelet examples, and you can definitely get the classes this way. The treatment in Anderson's Discrete Math would be typical - coloring an $$n$$-sided polygon with $$m$$ colors. Even with two colors and four sides becomes nontrivial and leads you to stabilizers etc. An interesting variant on this (which I don't recall where I first heard it) is to first talk about necklaces, then to talk about candles on birthday cakes, which of course can't be reflected so easily without a big mess on your table.

If you want something a little more interesting, though, I'd recommend trying music theory, such as "Why Are There Twenty-Nine Tetrachords", a nice article by Julian Hook. Here $$m=2$$ - either you have a note in a chord or you don't - and the rotation and reflection have very immediate rationales for certain genres of 'modern' music.

Or if you want something more advanced (perhaps using Pólya enumeration) there is a nice exercise in Anderson: "In how many ways can the squares on the top of a $$3\times 3$$ chessboard by colored?" That's a little more interesting than the square, but has the same symmetry. There are many helpful exercises in Keller/Trotter as well, such as "Xylene is an aromatic hydrocarbon having two methyl groups (and four hydrogen atoms) attached to the hexagonal carbon ring. How many isomers are there of xylene?"

With all these and more options, I figure you'll find something you find interesting enough to use as your base example, good luck!

• Organic chemistry was what I was going to say. Notably, it seems to be have been a major motivation for Pólya himself. – Peter Taylor Dec 18 '18 at 14:45

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.

I think this could be a motivational example.

True Story: My daughter and I tried to make a little card game once (when we were tired of the ones we owned), and discovered we needed Burnside's Lemma when it came to making the cards. No doubt this idea already exists somewhere and has a name, but we "invented" a game where players take turns laying square tiles (with colored edges) down on a board, where edges of neighboring tiles must match in color. When we allowed only two possible colors per edge, it was easy to make the cards and the deck was too small to make for much of a fun game. It was when we considered increasing the number of possible edge-colors and "possible" shapes for the tiles that we quickly lost our ability to even determine how many tiles there should be.

It was such a fun and natural way to stumble upon a need for a theorem, that I presented it to a group of undergrads (PowerPoint presentation), having them build and play the game before diving into the counting problem.

• Note: The animations in that PowerPoint only seem to work correctly when viewed in PP (not Google slides). – Nick C Dec 17 '18 at 23:27