We started today’s project by talking through the game. The question of whether the game ends in a finite number of steps was a little harder for them to understand than I think – but they got it eventually.

The confusion seemed to be the difference between:

(i) Here’s a path through the game than does end in a finite number of steps, and

(ii) there is no path that has an infinite number of steps.

After the short introduction to “Buckets of Fish” we watched Kelsey Houston-Edwards’s video on “Killing the Mathematical Hydra”. We’d watched this video when it came out a few weeks ago but didn’t use it for a project. I was happy that the boys remembered seeing it, though.

After viewing Houston-Edwards’s video we returned to whiteboard to talk about Hydras. My younger son did a nice job summarizing the hydra game, which I think is a testament to how good Houston-Edwards’s videos are explaining mathematical ideas to the public.

Next up – how are the hydras related to the fish?

Finally, to wrap up the project I thought it would be fun to study the case with 2 buckets more carefully. The motivation for this last section was a combination of the induction argument in Hamkins’s blog post and the introduction to induction proofs in Houston-Edwards’s latest video. For our project on that video see here: