Last month Mathologer published an incredible video on what he calls “Triangular Squares”:

I’ve been meaning to use this video for a project for the boys ever since I saw it. Today I finally got around to watching it with the boys.

Here are their initial thoughts after watching the video:

Now we went through some of the ideas. First I asked the boys to try to sketch Mathologer’s argument that is irrational. Then I asked what proof they would have given for that fact without seeing the video:

Next we explored the irrationality proof for :

Finally, we did a bit of exploration of the seeming paradox mentioned at the end of the video. That paradox is essentially -> the argument used to show that is irrational seems to also show that 3 times a triangular number can never be a triangular number. BUT, there are lots of examples showing that 3x a triangular number is a triangular number. What’s going on?

So, another terrific video from Mathologer. His ability to shed light on advanced math topics for the general public is incredible. I love using his videos to help my kids see amazing math ides from new and beautiful angles!