Today was our last day looking at Chapter 3 of Art of Problem Solving’s *Introduction to Counting and Probability* and I picked one of the challenge problems for our wrap up project. The problem has two parts and asks about counting arrangements of 10 people sitting around a table with a few constraints.

Here’s the boys approach to the first part of the problem where the constraint is that two people insist in sitting directly opposite from each other:

At the end of the last movie my older son had a question about reversing the position of the two people who insist on sitting opposite from each other. We spend a little extra time on that question because it is important to see that that switch is just a rotation of an arrangement that we’ve already counted.

The second piece of the challenge problem is that you now have two pairs of people who want to sit opposite each other. This problem was exactly at the right level for the kids – they couldn’t see the answer right away, but after talking about a few different approaches they were able to find their way to an answer in 5 minutes.

So, a nice little problem to wrap up the third chapter of our summer counting project. It is fun to hear kids talk through these counting problems. The problems are easy enough to understand, but also can be really challenging. I also like the way that all of the arithmetic inside of these counting problems help build up a little number sense, too.