# A pretty neat counting problem from Mathcounts

The boys came back from a 2 day camping trip today. I had some afternoon meetings, but luckily they got back early enough for us to do a little counting project.

Without anything in particular planned, I just picked the first problem in the challenge problem section at the end of chapter 2 in our Introduction to Counting and Probability book. The problem is a pretty neat case by case counting problem (even though it looks fairly dull at first glance):

How many positive integers between 24 and 125 have a digit sum that is a multiple of 7?

The boys did a nice job of breaking the problem up into cases right away. In the first video we look at the cases where we have a 2 digit integer. I really liked hearing my younger son talk about the two digit numbers whose digit sum is 14 – it is fun to hear a young kid bring together the ideas you need to work through this problem. One of the nice surprises about this problem is that there’s some interesting number sense ideas hiding inside of it!

In the next video we look at the three digit numbers from 100 to 125 whose digit sum is a multiple of 7. There’s a little surprise in this part, too, since there are no numbers in the range with a digit sum is 14. It takes a moment for the boys to realize that the largest digit sum isn’t necessarily the largest number, so that’s a another neat math idea hiding in this problem:

So a nice problem with a couple of neat ideas beyond the case by case counting. Fun little project to do in the lucky extra 15 minutes I had with the kids today.