But a proof isn’t an explanation it’s a proof. Proof by assuming the opposite is true and demonstrating a contradiction is very indirect and not at all directly explanatory yet it’s a proof non the less. The goal of proofs is to demonstrate something to be provably true, not expository knowledge gathering.

In fact most mathematicians (myself included!) think the more clever the trick the better the proof! The trick itself being clever is interesting because it often yields a new way of tackling or thinking about your own proofs. A bland explanatory proof that elicits some conceptually “why” is only preferable if it has a reason for doing so - does understanding why yield a new avenue of research? Often then the “why” is quite a clever trick too.

I think it’s a bit the opposite of programming. There you want your solutions to demonstrably not be clever and the code be its own documentation. It’s a different discipline.

Hmmm... seems to me that if you can find a solution without creating the desired explanation - then that's a problem with the original question - not the solution itself.

And discovering a bad question leads to the correct question. No?

Is this what the prompt means by proof strength gap, or is that something else entirely? (Sorry not a mathematician.)