I stumbled on this incredible presentation about tilings from Richard Stanley yesterday:
Richard Stanley’s presentation on Tilings
We’ve actually done many projects on tilings in the past if you search the blog – some of the fairly recent ones are:
Learning about tiling pentagons from Laura Taalman and Evelyn Lamb
Today we looked at two fun examples from Stanley’s paper -> tiling a chess board with dominoes and tiling a hexagon grid formed by the triangular numbers with “tribones.”
First up with the chess board. The problem here is pretty famous and a really fun one to try out with kids. Just in case you’ve not seen it before and want to try it out yourself, the problem is: If you remove two opposite corners of a chess board, can you tile the remaining shape with 2×1 dominoes:
Next we discussed the problem my younger son asked about – what happens if you removed two random squares of opposite color?
Now we moved on to the tribones and the hexagon grid. Here’s a quick discussion / introduction to the problem from Stanley’s paper:
Next I intended to have them try to build the T(9) shape from the tribones, but we took a little detour first to try to figure out why building T(6) from tribones was impossible. It probably took 10 minutes for the kids to find the argument, but it is was fun work. I wish I had left the camera running for it, but I didn’t. Here’s a short summary of the argument:
Finally, we wrapped up the project by trying to construct T(9) from the tribones:
So, a really fun weekend of tiling. I’m really happy that I stumbled on Stanley’s presentation yesterday!
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If you wanted the paper version of Stanley/Ardila, the wayback machine has a copy, http://web.archive.org/web/20070821175332/http://www.claymath.org/fas/senior_scholars/Stanley/tilings.pdf