What I hope was a productive struggle

My son encountered a really great, but pretty tough, problem from an old AMC 10 yesterday:

I mentioned that I’d really like to see James Tanton make a video about this problem because he has a fantastic way of presenting problems, and also his approach tends to be different than mine.

But, rather than a professional mathematician working through the problem, here are the thoughts my son had (with a little direction from me in the last two videos).

Part 1 – a struggle to understand what’s going on. This is about a 7 minute talk where there are some good ideas, but we don’t get on to a path that leads to the solution:

After the first video my son went to school. We picked up talking about the problem when he got home and essentially started over. I encouraged him to try to think of the problem in a new way. Because he mentioned some symmetry arguments in the number of ways to place “n” black squares on the board, I asked him to look for symmetry in the geometry in this problem.

After studying the geometry for a bit, for the last part of this project we tried to find a systematic way to look at the possible arrangement that solve the problem.

So, a great problem and hopefully a productive struggle. There are so many great ideas hiding in this problem – finding those ideas is one of the things that makes this problem so interesting and so challenging.