#** third hint is rot1

[notarealwelder]

Seriously though—

An amount of mathematics is...straightforward, for lack of a better word; for easier problems you can ~intuit* the proof, and as you go solving easier problems you very naturally develop a toolset to analyze and prove them, which transfers to harder problems. So you can start from a small* amount of ineffable intuition, definitions, and theorem-ideas, and grind your way into nontrivial skill levels of, I dunno, topology.

(This does not extend to the skill of, say, intuiting novel definitions or concepts, which is Important for research, but, ok, maybe this happens all by itself somehow.)

Many serious theorems you might not guess to be true offhand, or wouldn't guess to try to prove — there're many similar-looking statements, and this one is not obviously distinguishable as a provable theorem — but they're straightforward to discover: you can pretty much just play with definitions and concepts you have, apply them to each other in uncomplicated ways, and would you look at it, a theorem falls out. (Lagrange's theorem is like that; Sylow's likewise, though less obviously.)

(On the other hand, Lagrange's was noticed in 1771, but proven partially in 1801, and completely only in 1861, which...puts into perspective just how important good concepts/definitions are. Or how little time people had for math, or how much worse math-invention ability was propagated back then; I don't really know.)

But sometimes the whole theorem is basically a single bullshit insight, some trivial steps taking it to the conclusion, and absolutely no way to guess the approach from the shape of the problem.

Sure, you can consider frevrf sbez bs fva, wildly but very reasonably guess that it can be cerfragrq nf na vasvavgr cebqhpg of jut spput**, and from there the closed form of the sum of 1/n² is pretty obvious. But the question is: how do you know to look that way? Or how do you think to check / to try something in that vein?

In this particular case, the third hint seems unnecessary given the first two, but offhand I do not know the ~intuition that would lead one to guess them, nor how one can develop such an intuition.

It is dastardly frustrating.

(If somebody knows how to guess the first or the second hint, please tell me!)