This is also a much more difficult question and one that I wouldn’t expect the kids to be able to handle on their own. However, they did walk almost all the way to the solution by themselves, which was pretty cool.

We started by talking about the problem and discussing a few possible approaches to solving it – the conversation about understanding the problem was fantastic.

I was super happy with my younger son’s idea for checking a 3-4-5 triangle against a 4-5-6 triangle.

In the next part of the talk we explored the area of the 4-5-6 triangle using Heron’s formula. As we’ll see in a few videos, it was really lucky that my older son remembered this formula.

Also – ha ha – “decimals and square roots don’t usually work out” 🙂

At the end of the last video the boys wondered if both triangles could be right triangles. It took a while for them to see how to use the Pythagorean Theorem to approach this question, but eventually we got going. At the start it seems that we have a really complicated algebraic expression, but, almost by miracle, a parity argument appears!

Having had the luck of stumbling on a parity argument in the last video, I showed them how a similar parity argument would apply with Heron’s formula to show that there were no solutions to Tanton’s question. This is the one step that I don’t think they could have stumbled on by themselves. Even so, it was really great that Heron’s formula and a parity argument came up naturally while we were discussing the problem.

Finally, we wrapped up by discussing our thoughts about the rectangle and triangle problems:

So, thanks to Josh for pointing out this problem that I’d missed, and thanks to James Tanton for inspiring another really fun Family Math project.

## Comments

Fun problem! Here’s a desmos slider set of sides to get some solutions (non integer at least):

https://www.desmos.com/calculator/oa2v4enaer