What I’ve learned from this new hobby is the compliment: waffle-tastic!

Today, we did what I hoped would be waffle-tastic project – with waffles!

We started off counting squares on a waffle:

Once we counted the number of different ways to put one blueberry in the waffle, we looked at a new problem: what are the number of ways to put the blueberry in the waffle if we consider ways that are the same under rotation to be identical:

Having covered the ways of putting one blueberry on the waffle, the next part of the project was looking at two blueberries. First up was how many different ways could you put two blueberries onto the waffle. The interesting discussion here was about when two arrangements should be considered to be the same:

The last part of the project was a really difficult problem – how many ways are there to put two blueberries in the waffle when we consider ways that are the same under rotation to be identical?

I knew this would be a challenge, but it was harder for the boys to get their arms around than I expected. I split this discussion into two pieces – but the total discussion is still over 13 minutes.

I wish that I’d done a better job with this last part, but you not every one of these projects goes perfectly. Still a fun project overall, though, and maybe it opens the door to a future project counting symmetries.

Next time, consider encouraging them to test their counting techniques on smaller examples where they can list all the cases explicitly. In this case, considering a 2×2 and then a 3×3 waffle would probably have helped.

There are two identifications happening based on the symmetries of (a) swapping the blueberries and (b) rotating the waffle. This gives a diagram:
placements of 2 distinct fruits -> placements of 2 blueberries -> rotationally distinct placements of 2 blueberries. One frequent source of confusion is what level a count of something is being done. For example, the orbit 2 arrangements were basically counted at the top layer, but then applied at the middle layer.
Better language and notation to make these points clear would probably be helpful. As you can tell from the awkwardness with which I’ve phrased this, I don’t know the right language or notation.

I definitely want to revisit more Polya counting stuff. I totally misjudged the difficulty of this one, though, and then my attempt to help get us back on track totally backfired. Oh well . . . hope the next one goes better.

## Comments

Next time, consider encouraging them to test their counting techniques on smaller examples where they can list all the cases explicitly. In this case, considering a 2×2 and then a 3×3 waffle would probably have helped.

There are two identifications happening based on the symmetries of (a) swapping the blueberries and (b) rotating the waffle. This gives a diagram:

placements of 2 distinct fruits -> placements of 2 blueberries -> rotationally distinct placements of 2 blueberries. One frequent source of confusion is what level a count of something is being done. For example, the orbit 2 arrangements were basically counted at the top layer, but then applied at the middle layer.

Better language and notation to make these points clear would probably be helpful. As you can tell from the awkwardness with which I’ve phrased this, I don’t know the right language or notation.

I definitely want to revisit more Polya counting stuff. I totally misjudged the difficulty of this one, though, and then my attempt to help get us back on track totally backfired. Oh well . . . hope the next one goes better.