An introductory mathematical induction problem I learned thanks to Kelsey Houston-Edwards

Yesterday Annie Perkins asked about resources to help a student understand mathematical induction:

Hey #mtbos, I have a student doing a project on proof by induction and she needs help understanding the WHY of why it works. Anyone have solid resources for her?

(it's not being taught in my class currently, and I can't find anything solid beyond "how" right now)

— Annie Perkins (@anniek_p) January 18, 2020

Looking for some resources I was reminded of this incredible video that Kelsey Houston-Edwards made in 2017 – the proof by induction example starts around 5:25:

Today I thought it would be fun to have the boys go back and work through that problem. I started by introducing the problem after we’d made some L shapes out of snap cubes. The boys were able to understand the problem and solve the 1×1 and 2×2 cases:

Now we moved to the 4×4 case. My older son started by looking at different cases – which was a good way to solve the specific 4×4 case. At first it was hard to see how to generalize this approach, though my younger son had some nice ideas about how to extend it:

The next step was a deep dive into the 4×4 case and looked carefully at the exact problem we were trying to solve. Here the boys did a great job of seeing how to extend the solution of the 2×2 case to the 4×4 case.

Finally, we looked at how to extend the 4×4 case to the 8×8 case, and then how to extend to all cases of the problem. We ended the project by discussing how / why mathematical induction works on problems like this one: