Counting 2xN domino tilings

[note – not edited because of weekend driving-around-kid activities. Will edit when I get back this afternoon]

A while back we explored 2xN domino tilings at the suggestion of Jim Propp:

a fun counting exercise for kids suggested by Jim Propp

We just got back from vacation yesterday and I was looking for something easy to do this morning for our math project. Who knows why, but revisiting the 2xN tilings came to mind last night. The idea for today was as much about introductory ideas for proofs as it was for counting.

So, we started in by reviewing the problem and exploring the first few cases – the boys had a lot of interesting ideas about to count the cases:

Next we looked at the number of ways to tile a 2×4. From the work in the first video the predictions were either 4 or 5 ways. They found 5 and we had a short (non-rigorous) talk about why there weren’t more than 5:

Now we tackled the 2×5 case and were feeling pretty good about our guess that there would be 8 cases:

After all of the work getting used to the problem, we now moved on to see if we could understand where the Fibonacci pattern was coming from. Specifically, could we see why the number of different ways to tile the 2×5 rectangle was equal to the number of ways to tile the 2×4 plus the number of ways to tile the 2×3. This part of the project gets to some of the basic ideas about proof and presents some pretty fun challenges for kids:

We don’t quite find the correspondence in this video.

In the last video we didn’t find the Fibonacci correspondence that we were looking for, but refining some of the arguments from the last video actually did get us there.

After we finished with this piece I asked the boys to used the ideas from the Fibonacci argument to find all of the tilings for the 2×6 rectangle. They were able to find all of those tilings without having to start from scratch!

So, a fun and fairly light project this morning. Thinking about using the idea here for one of the Family Math nights at my son’s elementary school, though the number of dominoes required for a room full of kids might be prohibitive.