A challenging case by case counting problem

We were finishing up Chapter 2 of Art of Problem Solving’s Introduction to Counting and Probability this morning and ran across a pretty challenging counting problem. It made for an excellent little project with the boys since there are multiple ways to approach the problem.

Here’s a quick introduction to the problem:


First up was my younger son using the “b’s” as the cases. We’d already spent time going through this approach, but I wanted to make sure that the kids had understood the cases.


Next up was my older son counting using the “c’s” as cases. Both kids really struggled with this approach when we talked through the problem earlier in the morning, and my son struggled a little to get going here. Once he understood that the case we were looking at first was the case c = 2, though, he was able to work through the problem.


In the last video we didn’t quite get to the end after 5 minutes, so we broke the discussion into two pieces. Here we finished up the “c” case and found that our answer matches the answer we got when we looked at the cases involving the “b’s”. Adding up the various “c” cases was a little challenging for the boys the first time around, so that was another reason that I wanted to go slowly for this last part of the project.


So, although a fairly contrived counting problem, I like this project a lot. The kids got some nice practice working with both algebraic expressions and with numbers. They also had to understand a little bit about inequalities to work thorough the problem. Of course, from the point of view of learning about counting techniques, the nice thing about this problem is that there are multiple ways to break the problem into cases. Seeing that those to approaches lead to the same answer hopefully helps them gain a better understanding of case by case counting.