Unexpected appearances of $\pi^2 /~6$. 
"The number $\frac 16 \pi^2$ turns up surprisingly often and frequently in unexpected places." - Julian Havil, Gamma: Exploring Euler's Constant.


It is well-known, especially in 'pop math,' that
$$\zeta(2)=\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\cdots = \frac{\pi^2}{6}.$$
Euler's proof of which is nice. I would like to know where else this constant appears non-trivially. This is a bit broad, so here are the specifics of my question:

*

*We can fiddle with the zeta function at arbitrary even integer values to eek out a $\zeta(2)$. I would consider these 'appearances' of $\frac 16 \pi^2$ to be redundant and ask that they not be mentioned unless you have some wickedly compelling reason to include it.

*By 'non-trivially,' I mean that I do not want converging series, integrals, etc. where it is obvious that $c\pi$ or $c\pi^2$ with $c \in \mathbb{Q}$ can simply be 'factored out' in some way such that it looks like $c\pi^2$ was included after-the-fact so that said series, integral, etc. would equal $\frac 16 \pi^2$. For instance, $\sum \frac{\pi^2}{6\cdot2^n} = \frac 16 \pi^2$, but clearly the appearance of $\frac 16\pi^2$ here is contrived. (But, if you have an answer that seems very interesting but you're unsure if it fits the 'non-trivial' bill, keep in mind that nobody will actually stop you from posting it.)

I hope this is specific enough. This was my attempt at formally saying 'I want to see all the interesting ways we can make $\frac 16 \pi^2$.' With all that being said, I will give my favorite example as an answer below! :$)$

There used to be a chunk of text explaining why this question should be reopened here. It was reopened, so I removed it.
 A: Related, but certainly not in an immediately obvious way, to $\zeta(2)$ is the density of the squarefree numbers.
Call a natural number squarefree if no square larger than $1$ divides it (e.g. 12 is not squarefree because 4 divides it, but 30 is squarefree). Let $S$ be the set of squarefree numbers. Then
$$
    \lim_{n\to\infty}\frac{\#([1..n]\cap S)}{n} = \frac{6}{\pi^2}.
$$
See here: http://mathworld.wolfram.com/Squarefree.html
A: I'll give you three cute examples from the book, (Almost) Impossible Integrals, Sums, and Series.

A particular case of the generalization from Section $1.11$, page $7$
$$i)\  1- \int_0^1 \left(2 x+ 2^2 x^{2^2-1}+2^3 x^{2^3-1}+2^4 x^{2^4-1}+\cdots\right) \frac{\log(x)}{1+x} \textrm{d}x=\frac{\pi^2}{6}.$$
A particular case of the generalization from Section $1.38$, page $25$
$$ii) \ \frac{1}{2}\int_0^{ \infty} \int_0^{\infty}\frac{x -y}{e^x-e^y} \textrm{d}x  \textrm{d}y=\int_0^{ \infty} \int_0^y\frac{x -y}{e^x-e^y} \textrm{d}x  \textrm{d}y=\frac{\pi^2}{6}.$$
The first example from Section $1.17$, page $10$
$$\frac{6}{7\zeta(3)}\int _0^1 \int _0^1 \frac{\displaystyle \log \left(\frac{1}{x}\right)-\log \left( \frac{1}{y} \right)}{\displaystyle \log \left(\log \left(\frac{1}{x}\right)\right)-\log \left(\log \left(\frac{1}{y}\right)\right)} \textrm{d}x \textrm{d}y =\frac{6}{\pi^2}.$$

Another curious sum of (crazy) integrals leading to the same value which was proposed by the author of the mentioned book is

$$\frac{\pi^2}{6}=\frac{4}{3}\int_0^{\pi/2}  \log \left(\frac{\left(x^2\sin^2(x)+ \pi ^2/4 \cos ^2(x)\right)^{x/2}}{x^x}\right)\sec ^2(x) \textrm{d}x$$
$$-\frac{2}{3} \int_0^1 \frac{\log \left(\left(x^2+\left(1-x^2\right) \cos (\pi  x)+1\right)/2\right)}{x-x^3} \textrm{d}x,$$

but also the amazing $\zeta(2)\zeta(3)$ product in the harmonic series (with zeta tail) form

$$\frac{1}{2\zeta(3)}\sum_{n=1}^{\infty} \frac{H_n^2}{n}\left(\zeta(2)-1-\frac{1}{2^2}-\cdots-\frac{1}{n^2}\right)=\frac{\pi^2}{6},$$

or 

$$\frac{\pi^2}{6}=4\sum_{n=1}^{\infty}\biggr(2n\biggr(1-\frac{1}{2^{2n+1}}\biggr)\zeta(2n+1)-2\log(2)\biggr(1-\frac{1}{2^{2n}}\biggr)\zeta(2n)$$
$$-\frac{1}{2^{2n}}\sum_{k=1}^{n-1}(1-2^{k+1})\zeta(k+1)(1-2^{2n-k})\zeta(2n-k)\biggr).$$

A: Consider the following picture:, centered at the origin of $\mathbf{R}^{2}.$ It is a concentric arrangement of circles $\color{red}{\text{(- this should be discs ?)}}$; each circle has radius $1/n.$ We can think of it as an infinite bulls-eye. The sum of the areas shaded in red is equal to $\frac{1}{2}\pi\zeta(2).$ In particular
$$\sum_{k=1}^\infty \int_0^1 \frac{\sin (\pi (2k - 1)/ r)}{2k - 1} r \, dr = \frac{\pi}{8}\left(1-\zeta(2)\right),
$$
Surprisingly if you take this arrangement and rotate it about the $x-$axis then you have a similar arrangement with circles being replaced by $3-$balls each with radius $1/n.$ In this case the sum of volumes shaded in red is equal to $\pi\zeta(3).$ 

Update: It dawned on me that I can in fact extend the notion "volume shaded in red" to higher dimensions. 
Let $K_{i}$ be the $n-$ ball at the center of the origin of Euclidean $n-$space, $\mathbf{E}^{n},$ with radius $\frac{1}{i}$ and whose volume I denote by $\mu\left(K_{i}\right).$ Consider
$$
\sum\limits_{i=1}^{\infty}(-1)^{i+1}\mu\left(K_{i}\right).
$$
A closed form for this quantity is known whenever $n$ is an even number:
$$
(-1)^{1+\frac{n}{2}}{\left(2^{n-1}-1 \right)B_{n}\above 1.5pt \Gamma(1+\frac{n}{2})\Gamma(1+n)}\pi^{\frac{3}{2}n}.
$$ 
Inspections shows the numerator of the rational part is the sequence A036280(n/2). You can check in the case that $n=2$ the quantity computes to $\frac{1}{2}\pi\zeta(2).$
A: Here is a crazy-looking integral, which I believe I originally saw on the (now abandoned) integrals and series forum:
$$\int_{0}^{1}\frac{1}{\sqrt{1-x^2}}\arctan\left(\frac{88\sqrt{21}}{36x^2+215}\right)dx=\frac{\pi^2}{6}$$
A: A somewhat surprising occurence, which can be seen immediatelly via the Euler product, appears in the study of visible points of lattice.
Given a lattice $\Gamma \subset \mathbb R^d$, meaning $\Gamma=\mathbb Z v_1 \oplus ... \oplus\mathbb Z v_d$ for some $\mathbb R$ basis $v_1,.., v_d$ of $\mathbb R^d$, the visible points of $\Gamma$ are defined as 
$$V:= \{ z=n_1v_1+...+n_dv_d : n_1,.., n_d \in \mathbb Z , \mbox{ gcd } (n_1,.., n_d)=1 \}$$
The, we have the following result, (see Prop.~6 in Diffraction from visible lattice points and k-th power free integers)
Proposition The natural density of $V$ is 
$$
\mbox{dens}(V)=\frac{1}{ \det(A) \zeta(d) }
$$
where $A$ is the matrix with columns $v_1,v_2,...,v_d$. Here, natural density means the density calculated with respect to the sequence $A_n=[-n,n]^d$, note that this set can have a different density with respect to other sequences.
In particular, the visible sets of $\mathbb Z^2$, given by 
$$V=\{ (n,m) \in \mathbb Z^2 : \mbox{gcd}(m,n) =1 \}$$
have natural density $\frac{1}{\zeta(2)}$.

The so called "cut and project" formalism establises a connection between the above example and some sets in compact groups, which appeared in my research area recently.
Consider the group 
$$\mathbb K:= \prod_{p \in P} \left( \mathbb Z^2 / p \mathbb Z^2 \right)$$
where $p$ denotes the set of all primes. $\mathbb K$ is a compact Abelian group, and hence has a probability Haar measure $\theta_{\mathbb K}$.
Now, $\phi(m,n) := \left( (m,n)+p \mathbb Z^2 \right)_{p \in P}$ defines an embedding of $\mathbb Z^2$ into $\mathbb K$.
Define the set $$W:=   \prod_{p \in P} \left( \bigl(\mathbb Z^2 / p \mathbb Z^2\bigr) \backslash \{ (0,0) + p\mathbb Z^2 \} \right)$$
Then, the visible points of $\mathbb Z^2$ are exactly 
$$V= \phi^{-1}(W)$$
The set $W$, which is used in the study of diffraction of $V$, has the following properties:


*

*$W$ is closed and hence compact.

*$W$ has empty interior (hence is fractal shape).

*$\theta_{\mathbb K}(W) = \frac{1}{\zeta(2)}$
The last property is where I was going to, and it is intuitively not that hard to see once you identify $\theta_{\mathbb K}(W)$ as the product of the counting measures on $\mathbb Z^2 / p \mathbb Z^2$: this immediatelly gives 
$$\theta_{\mathbb K}(W) = \prod_{p \in P}\frac{p^2-1}{p^2}$$
P.S. There are similar appearences of $\zeta(n)$ in the study of $k$th power free integers, that is all the integers $n \in \mathbb Z$ which are not divisible by the $k$th power of any prime, for a fixed $k$.
A: Two simple trigonometric integrals are
$$\frac{4}{3}\int_0^1 \frac{\arctan{x}}{1+x^2}dx =\frac{\pi^2}{6}$$ and
$$\frac{4}{3}\int_0^1 \frac{\arcsin{x}}{\sqrt{1-x^2}}dx = \frac{\pi^2}{6}$$

Using inverse hyperbolic functions:
$$\frac{10}{3} \int_0^1\frac{\operatorname{arcsinh}\left({\frac{x}{2}}\right)}{x}dx=\frac{\pi^2}{6}$$
$$\frac{4}{3} \int_0^1 \frac{\operatorname{arctanh}{x}}{x} dx = \frac{\pi^2}{6}$$

From series
$$ \sum_{k=0}^\infty \frac{1}{((k+1)(k+2))^2} = \frac{\pi^2}{3}-3 $$
and
$$ \sum_{k=0}^\infty \frac{k}{((k+1)(k+2))^2} = 5- \frac{\pi^2}{2} $$
$\frac{\pi^2}{6}$ arises directly when cancelling out the integer terms:
$$ \sum_{k=0}^\infty \frac{5+3k}{((k+1)(k+2))^2} =\frac{\pi^2}{6}$$
Similarly, 
$$\frac{8}{3}\sum_{k=0}^\infty \frac{4k+5}{(2k+1)^2(2k+3)^2} = \frac{\pi^2}{6}$$

More series and integrals are available at http://oeis.org/A013661
A: This paper gives a polynomial-time approximation algorithm for the Minimum Equivalent Digraph (MEG) problem, with approximation ratio $\pi^2/6$.  
The problem is, given a directed graph, to find a min-size subset $S$ of the edges that preserves all reachability relations between pairs of vertices.  (That is, for every pair $u, v$ of vertices, if there is a path from $u$ to $v$ in the original graph, then there is such a path that uses only edges in $S$.)  The problem is NP-hard.  This was the first poly-time algorithm with approximation ratio less than 2.
A: Integral representations are given by
$$2\int_0^1 x \left \lfloor{\frac1x}\right \rfloor \ dx=2\int_1^\infty\frac{1}{t}\lfloor{t\rfloor}\frac{dt}{t^2}=2\sum_{n=1}^{\infty}\int_n^{\infty}t^{-3}\,dt=\sum_{n=1}^{\infty}n^{-2}=\frac{\pi^2}{6}$$
and
$$\frac{4}{3}\int_0^1\int_0^1\frac{1}{1-x^2y^2}\,dxdy=\frac{\pi^2}{6}$$
Then, for every $-1 < \alpha \le 1$,
$$\int_0^\infty\frac{\log(1+\alpha x)}{x(1+x)}\,dx= \log(\alpha)\log(1-\alpha)+\text{Li}_2(\alpha)$$
and when $\alpha=1$, this integral becomes
$$\int_0^\infty\frac{\log(1+ x)}{x(1+x)}\,dx= \frac{\pi^2}{6}$$
A: This I found neat
$$\int_0^\pi \frac{\log(\frac{\cos x}{2}+1)}{\cos x} dx=\frac{\pi^2}{6}$$
A: This is one of those amazing series for $1/\pi^2$. You can find them in this paper by G. Almkvist and J. Guillera.
$$\left(\frac{2}{5}\right)^{3}\sum_{n=0}^{\infty}\frac{(6n)!}{n!^{6}10^{6n}}(532n^{2}+126n+9)=\frac{6}{\pi^{2}}$$
A: Here is one,
$$-\sum_{n=0}^{\infty}\left[\zeta(2n)-\zeta(2n+2)-\zeta(2n+3)+\zeta(2n+4)\right]=\frac{\pi^2}{6}$$
A: How about
$$\int_0^1 dx \, \log{x} \, \log{(1-x)} = 2 - \frac{\pi^2}{6} $$
A: Let $I(n)$ be the probability that two integers chosen randomly from $[1,n]$ are coprime. Then, 
$$\lim_{n \to \infty} I(n)=\frac{6}{\pi^2}.$$
So, you could say the odds that two randomly-chosen positive integers are coprime is $1$ in $\frac{\pi^2}6$.
A: For $0< x<1$,
\begin{align}\text{Li}_2(x)+\text{Li}_2(1-x)+\ln x\ln(1-x)=\dfrac{\pi^2}{6}\end{align}
And, for $0\leq x\leq 1$, $\displaystyle \text{Li}_2(x)=\sum_{n=1}^\infty \dfrac{x^n}{n^2}$
A: In physics, $\pi^2/6$ appears as a proportionality constant between a metal's internal energy (or at least the contribution of the electrons to that energy) on the one hand and the density of states $\times$ the Fermi temperature on the other hand. It appears there as another manifestation of the identity $\zeta(2)=\pi^2/6$, i.e. its derivation has not really an independent character here compared to the installments of the other answers.
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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With Abel-Plana Formula:
\begin{align}
{\pi^{2} \over 6\phantom{^{2}}} & =
{3 \over 2} + 2\int_{0}^{\infty}{\sin\pars{2\arctan\pars{t}} \over
\pars{1 + t^{2}}\pars{\expo{2\pi{\large t}} - 1}}\,\dd t
\end{align}
A: I was quite pleasantly surprised to derive the continued fraction identity
$$\frac{\pi^2}{6}=-1+\cfrac{1}{1-\cfrac{1}{2-\cfrac{1}{5-\cfrac{16}{13-\cfrac{81}{25-\cfrac{256}{41-\cdots}}}}}},$$
which comes from the identity for the polylogarithm $\mathrm{Li}_s(z)$:
$$\mathrm{Li}_s(z)+1=\cfrac{1}{1-\cfrac{z}{1+z-\cfrac{z}{2^s+z-\cfrac{2^{2s}z}{3^s+2^sz-\cfrac{3^{2s}z}{4^s+3^sz-\cfrac{4^{2s}z}{5^s+4^sz-\cdots}}}}}},$$
which is valid for $\Re(s)>1$ and $|z|\le1$.
A: Define a continuous analog of the binomial coefficient as
$$\binom{x}{y}=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}.$$
While exploring integrals of the form
$$\int_{-\infty}^\infty\prod_{n=1}^m\binom{x_n}{t}\,\mathrm dt$$
I was surprised the first time I saw
$$\int_{-\infty}^\infty\binom{1}{t}^3\,\mathrm dt=\frac{3}{2}+\frac{6}{\pi^2}$$
show up.
A: Unexpected at first glance is $$2\sum_{m\ge1}\sum_{n\ge1}\frac{(-1)^n}{n^3}\sin(\tfrac{n}{m^{2k}})=\frac{1}{6}\zeta(6k)-\frac{\pi^2}{6}\zeta(2k).$$
A generalization may be found here.
Perhaps more unexpected is $$\sqrt3 \int_0^\infty \frac{\arctan x}{x^2+x+1} \, dx=\frac{\pi^2}{6},$$
which is proven here.
Even nicer is $$\frac1{12}\int_0^{2\pi}\frac{x\,dx}{\phi-\cos^2 x}=\frac{\pi^2}6,$$
which can be seen here. Here $\phi=\frac{1+\sqrt5}2$ is the golden ratio.
A pleasing logarithmic integral is $$\frac83\int_1^{1+\sqrt2}\frac{\ln x}{x^2-1}dx=\frac{\pi^2}6-\frac23\ln^2(1+\sqrt2),$$
proven here.
Another nice trigonometric integral:
$$2\int_0^{\pi/2}\cot^{-1}\sqrt{1+\csc x}\, dx=\frac{\pi^2}{6},$$
from here.

Edit: as was stated in the comments of this answer, it's the $\pi^2$ that counts, though un-scaled integrals evaluating to $\pi^2/6$ are best. With this in mind, I present a nice $\zeta$-quotient integral involving $\pi^2$:
$$\int_0^\infty \left(\frac{\tanh(x)}{x^3}-\frac{1}{x^2\cosh^2(x)}\right)\, dx=\frac{7\zeta(3)}{\pi^2}=\frac{7\zeta(3)}{6\zeta(2)},$$
shown here.

I just derived another identity:
$$\int_0^\infty\frac{(x^2+1)\arctan x}{x^4+\tfrac14x^2+1}dx=\frac{\pi^2}{6}.$$
Since I just found this identity I present the proof. In the link I provided after the second identity it is shown that 
$$f(a)=\int_0^\infty \frac{\arctan x}{x^2+2ax+1}dx=\frac{\pi}{4\sqrt{1-a^2}}\left(\frac\pi2-\phi(a)\right)\qquad |a|<1$$
where $\phi(a)=\arctan\frac{a}{\sqrt{1-a^2}}$. First off, notice that $\phi(-a)=-\phi(a)$. Thus 
$$j(a)=\frac12(f(a)+f(-a))=\int_0^\infty\frac{(x^2+1)\arctan x}{x^4+2(1-2a^2)x^2+1}dx=\frac{\pi^2}{8\sqrt{1-a^2}}.$$
Hence $$j(\sqrt{7}/4)=\int_0^\infty\frac{(x^2+1)\arctan x}{x^4+\tfrac14x^2+1}dx=\frac{\pi^2}{6}.$$

Expect more nice examples as I gather the best ones.
A: Draw line segments from the centre of a unit circle, to two uniformly random points on the circle.

The expected product of the areas is $\frac{1}{\pi}\int_0^\pi\frac{1}{2}x\left(\pi-\frac{1}{2}x\right)dx=\frac{\pi^2}{6}$.
This question asks whether this result can be used to solve the Basel problem.
A: Problem 11953 from AMM (January 2017) asked for the evaluation of the following double integral whose value turns out to be equal to $\frac{\pi^2}{6}$.
$$\int_0^\infty \!\!\!\int_0^\infty \frac{\sin x \sin y \sin (x + y)}{xy(x + y)} \, dx \, dy = \frac{\pi^2}{6}.$$
Problem 2074 from Mathematics Magazine (June 2019) asked for the following evaluation of a limit of a sum whose value turns out to be equal to $\frac{\pi^2}{6}$.
$$\lim_{n \to \infty} \sum_{k = 1}^n \frac{(-1)^{k + 1}}{k} \binom{n}{k} H_k = \frac{\pi^2}{6}.$$
Here $H_n = \sum_{k = 1}^n \frac{1}{k}$ denotes the $n$th Harmonic number.
And here are a few sums:
$$\sum_{n = 1}^\infty \frac{H_n}{n2^{n - 1}} = \frac{\pi^2}{6}.$$
$$\sum_{n = 1}^\infty \frac{H_n}{n (n + 1)} = \frac{\pi^2}{6}.$$
$$\frac{3}{2} \sum_{n = 0}^\infty \left (\frac{1}{(6n + 1)^2} + \frac{1}{(6n + 5)^2} \right ) = \frac{\pi^2}{6}.$$
and
$$\sum_{n = 1}^\infty \frac{3}{n^2 \binom{2n}{n}} = \frac{\pi^2}{6}.$$
And a few more sums, this time involving the variant harmonic number term $\Lambda_n$ were
$$\Lambda_n = 1 + \frac{1}{3} + \cdots + \frac{1}{2n - 1} = \sum_{k = 1}^n \frac{1}{2k - 1} = H_{2n} - \frac{1}{2} H_n.$$
$$\sum_{n = 1}^\infty \frac{\Lambda_n}{n(2n - 1)} = \frac{\pi^2}{6},$$
$$\sum_{n = 1}^\infty (-1)^{n+ 1} \left (\frac{2n + 1}{n(n+ 1)} \right )^2 \Lambda_n = \frac{\pi^2}{6},$$
and
$$2\sum_{n = 1}^\infty \frac{(-1)^{n + 1} \Lambda_n}{3^{n - 1} n} = \frac{\pi^2}{6}.$$
Some function values:
$$\zeta (2) = \operatorname{Li}_2 (1) = \frac{\pi^2}{6},$$
where $\zeta$ denotes the Riemann zeta function while $\operatorname{Li}_2 (x)$ is the dilogarithm.
$$6 \operatorname{Li}_2 \left (\frac{1}{2} \right ) - 6 \operatorname{Li}_2 \left (\frac{1}{4} \right ) - 2 \operatorname{Li}_2 \left (\frac{1}{8} \right ) + \operatorname{Li}_2 \left (\frac{1}{64} \right ) = \frac{\pi^2}{6}.$$
And some strange integrals:
$$\int_0^1 (x^{-x})^{{{(x^{-x})}^{(x^{-x})}}^\cdots} \, dx = \frac{\pi^2}{6},$$
and
$$\int_0^\infty \frac{dx}{\operatorname{Ai}^2 (x) + \operatorname{Bi}^2(x)} = \frac{\pi^2}{6},$$
where $\operatorname{Ai}(x)$ and $\operatorname{Bi}(x)$ denote the Airy functions of the first and second kinds, respectively.
A: Here is the simplest,
$$\int_0^1 \frac{\ln x}{x-1}\ \mathrm{d}x=\frac{\pi^2}6$$
A: In terms of the two real branches of 
the Lambert W function
$\require{begingroup} \begingroup$
$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$
\begin{align}
\int_0^1
\frac{\Wp(-\tfrac t{\mathrm e})\,(\Wp(-\tfrac t{\mathrm e})-\Wm(-\tfrac t{\mathrm e}))}
{t\,(1+\Wp(-\tfrac t{\mathrm e}))\,(1+\Wm(-\tfrac t{\mathrm e}))}\, dt
&=\frac{\pi^2}6
\tag{1}\label{1}
.
\end{align} 

Edit
And another one, with different integrand curve:
\begin{align} 
\int_0^1 
\frac{\Wp(-\tfrac t\e)+t\,(1+\Wm(-\tfrac t\e))}{t\,(1+\Wm(-\tfrac t\e))}
\, dt
&=\frac{\pi^2}6
\tag{2}\label{2}
.
\end{align} 
The intersection point of the 
integrands 
in \eqref{1} and \eqref{2}
can be found exactly at
$t=\tfrac1\Omega-1\approx 0.763222834$,
where $\Omega$ is 
Omega constant,
$\Omega \e^{\Omega }=1,\ \Omega=\W(1)\approx 0.56714329$ 
(thanks, @omegadot).
Also, one more:
\begin{align} 
\int_0^1 \ln\left(\frac{-\Wm(-t\,\exp(-t))}t\right) \, dt
&=
\int_0^1 -t-\Wm(-t\,\exp(-t)) \, dt
=\frac{\pi^2}6
\tag{3}\label{3}
.
\end{align}
$\endgroup$
A: You have $$\int\limits_{-\infty}^{+\infty}\int\limits_{-\infty}^{+\infty}\frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\frac{\pi^2}{2}.$$ Of course, dividing by 3 you have the expected value $\pi^2/6$.
A: $$\sum_{n=0}^{\infty}\frac{C_n^5}{2^{10n}}\left(\frac{n+1}{2n-1}\right)^4(4n-1)\sum_{k=0}^{n}(-1)^k{n \choose k}\frac{(n-k)[4k^2-2k+1]}{(2k-1)(2k+1)}=\frac{4}{\pi^2}$$
Where $C_n$; Catalan numbers
A: A regular $n$-gon of side length $n$ is inscribed in a circle. As $n\to\infty$, the difference between their perimeters approaches $\frac{\pi^2}{6}$.
A: Another natural integral evaluates exactly to $\frac{\pi^2}6$
$$\int_0^\infty \frac{x^2}{\sinh^2 x}dx=\frac{\pi^2}6$$
A: Start with 1 as your initial score, and one white ball with one black ball in a bag. You pick up at random one of them, and you do it twice (independently). If you get white in both cases, then you advance. Otherwise you stop and your score is 1. Everytime you advance, 1 is added up to your score and one new white ball is added into the bag, and you proceed as before, and so on, until you have to stop. The expected value of your final score is $\zeta(2)$.
I know, we don't get $\pi^2/6$ as requested, but I find it interesting that this number (anyhow we write it) is the expected value of the score in a game so easy to play.
