[had to publish without editing, sorry – a little time constrained today]

A seemingly innocent problem in Art of Problem Solving’s Introduction to Geometry sparked a fun conversation about counting and symmetry this morning. It was such an interesting topic, and little background in geometry was required, so I asked my younger son to join in, too.

The original problem was this – If a regular polygon has 27 diagonals, how many sides does it have?

We started by talking about connecting points in a plane. Sorry for the blurry start at the beginning of this video, I wasn’t paying attention to the camera – only lasts about 10 seconds 🙂

It was interesting to me to hear how the boys counted the inside and outside line segments in the picture we drew here.

In the second step we looked at the number of line segments that you would need to connect “n” dots. Other than an odd comment about a factorial, it seemed like the boys caught on to the counting process. I tried to connect the formula we got here to adding up 1 + 2 + 3 + . . . + (n-1), but the kids didn’t see it, so I moved back to the geometry.

In the last part we returned to the original problem – how many diagonals are there? My younger son picked up on the (n-3) line segments that were diagonal, which was nice.

I also liked hearing their approach to solving (n)(n-3) = 54 at the end.

So, a nice little project coming out of a review problem in our geometry book. I like opportunities like this to show the kids a little more math than just the problem they were working on.