I learnt Cell 2 as the first proof by contradiction presented at University.
I believe that I came across the naive sum in Cell 3 on Richard Perring’s (@LearningMaths) twitter feed.
I’m not sure if, how and when I might insert any of these methods into the curriculum. I’m quite sure that I would develop exploratory tasks, possibly to deepen and stretch the knowledge of higher attaining learners. The prompt acts as a reminder that there are always creative and interesting avenues to solve even the most familiar of problems.
Additional links: for cell 3:
1. Here is a geogebra applet I made to visualise the naive sum.
2. A lovely little proof of the naive sum by Matt Parker (@standupmaths).
Dan Pearcy 