# A challenging counting example from Art of Problem Solving

We finished up the last section of chapter 3 in Art of Problem Solving’s Introduction to Counting and Probability today. The section is an introduction to counting with symmetry. The topic was one of the most eye-opening topics for me as an undergraduate, and some of the advanced ideas – Polya’s theory of counting, for example – are among the most beautiful ideas I ever saw in math.

For our project today we reviewed one of the exercises from the end of the chapter. The problem deals with arranging two groups of 5 people around a table. In the book the two groups are democrats and republicans, but we went with lego figures instead. Our groups are “nindroids” and “flood infection spores” . . . .

Part 1: How many distinct ways are there to arrange the 10 people if there’s no restrictions in the seating arrangement, but we’ll say that two arrangements are the same if they differ only by rotation?

[note: sorry for having 6 flood infection spores in the video – didn’t notice this problem until we started in on the next video.]

Part 2: In this next case we’ll add the seating restriction that each of the two groups of 5 people must sit together. My attempt to provide a “better” explanation of how to think about this problem was a disaster – sorry about that!

Part 3: The last case we looked at was the number of ways for the two groups to sit in an alternating pattern. This version of the problem caused a lot of difficulty for the boys when we were going through it the first time. Having them think through and explain this problem a second time was the main motivation for this project:

So, maybe not our most error free project, but still a fun one. I think that these “counting with symmetry” problems are great ways for kids to see some math that is both fun and challenging. These problems also show that counting arrangements can be a little more difficult that it might initially seem.