Prove that the sum of the reciprocals of the squares is π²/6

Prove that
$$\sum _{n=1}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}.$$

Posed by Pietro Mengoli in 1650. Jakob Bernoulli could prove the series converges and is less than $2$, and then wrote, in print, that he could not find its value and would be grateful to anyone who could. It stood for ninety years. Euler settled it in 1735 and it made his name across Europe.

The difficulty is not in believing the answer — you can confirm it numerically in one line. The difficulty is that $\pi$ has no business being here. Nothing in the statement mentions a circle, an angle, or a curve of any kind.

A proof that does not say where the $\pi$ comes from has not really finished.

One warning, since it is the argument most people reach for first. Euler's original method — factoring $\sin x$ as an infinite product over its roots and comparing coefficients — is not rigorous as he gave it. He treated an infinite series as though it were a polynomial, and the same move produces false answers elsewhere. It can be made rigorous, but that is work, not a footnote. Say plainly whether what you are posting is complete or a sketch.

Open. No proof has been posted yet.