The cool thing in the 2nd solution is that it uses intuition about the question

the answer is the same regardless of r1's value, and will remain the same even as it diminishes to 0

It's an interesting way to look at problems in general...of course you can't always assume that what you think you see or don't see in the question is intentional.

Ah too bad - you publish any of these, by chance?

I've heard the variant of this where there's one ball that's heavier, but that's muuuch easier.

Fair enough - I'm an engineer...I know just enough math to be dangerously wrong sometimes. You might enjoy this one if you haven't seen it already.

Very interesting proof. Are there any other patterns like this that exist among square (or nth powers in general)?

Good call applying Bayes' Theorem that way. My first thought was to use the HH information just to eliminate the all-tails coin. I think your logic is valid.