A challenging counting problem from the 2011 AMC 8

My older son has been preparing for the AMC8 and this problem from 2011 gave him a little trouble:

 

We talked through it this morning and he was still a little confused about why his original answer isn’t correct. The error is pretty subtle – especially for a test that gives you roughly 2 min per problem:

So, we talked for a bit more and he was able to find some numbers that he counted that did not fit the requirements of the problem:

Finally, when he got home from school tonight we revisited the problem and counted the number of solutions directly:

I like this problem a lot – it is a great one for helping you learn how to count carefully!