This problem from the 2015 AMC 8 gave my son some trouble today. Actually quite a bit of trouble:

I’m not 100% sure what caused the difficulty. It might be that once you start thinking about this problem one way that it is hard to switch. Whatever the cause, though, we had a really good conversation about the problem.

Here’s his original approach that is incorrect:

So, after finding out that the answer in the last video was incorrect, we went to try to find the error. He found it pretty quickly.

After that I tried to explain an alternate approach to the problem. Unfortunately my explanation ended up causing quite a bit of confusion:

In the last part of our discussion I tried to dig my way out of the hole I created in the last video.

Even watching this video after the fact, I’m not sure what was the original source of his confusion. There was definitely some difficulty going from 4 parallel edges to 6 pairs of parallel edges.

By the end of our conversation he was able to walk through the argument, but I think that I’ll revisit some similar problems with him just to be sure the main ideas have sunk in.

I think this short project is a nice example of how old contest problem can help kids learn math. For me anyway, it is really challenging to come up with good problems and the fact that all of the old AMC problems are available for kids to work through is an incredibly helpful resource. Hopefully I can find some similar counting problems on other old AMC contests.