A couple of years ago Jim Propp suggested a neat counting exercise for the boys – counting tilings of 2xN rectangles by 2×1 dominos. We’ve played with this idea twice before, but thought it would be fun to revisit it today.

I started by reminding the boys of the problem and we checked out a few of the simple cases:

Next we looked at the 2×4 case and found that there were 5 possible arrangements:

Now we moved on to the 2×5 case – we had a long discussion about how to determine if we had found all of the possibilities:

Finally, we discussed why the Fibonacci pattern we were finding was correct. The argument here is a slightly sophisticated one for kids, but they were able to find it!