**Motivation.** As I was playing the pairs-matching game "Memory" (known as "Concentration" in some parts of the world) with my children, I was surprised that even thorough shuffling could not prevent quite a few pairs of cards lying next to each other. This inspired the following problem.

**Formalization.** If $X,Y$ are nonempty sets, let $\pi_0:(X\times Y)\to X$ be the map defined by $(x,y)\mapsto x$ for all $(x,y)\in X\times Y$. We formalize the "Memory" game by letting $n\in\mathbb{N}$ be a positive integer and looking at bijections $\varphi:\{1,\ldots, 2n\} \to \big(\{1,\dots,n\}\times\{0,1\}\big)$, and we think of $(i, 0), (i,1)$ as a "matching pair of cards" as in the "Memory" game. We denote the collection of such bijections by ${\cal M}_n$. We say $k\in \{1,\ldots, 2n-1\}$ is an *adjacent index* if $$\pi_0(\varphi(k)) = \pi_0(\varphi(k+1)),$$ and let $\text{Adj}(\varphi)$ be the collection of adjacent indices of the bijection $\varphi\in{\cal M}_n$. The expected value of the number of adjacent indexes in any such bijection $\varphi$ is given by $$E_n=\frac{1}{(2n)!}\sum_{\varphi\in{\cal M}_n} \text{Adj}(\varphi).$$

**Question.** What is the value of $\lim\sup_{n\to\infty} E_n$?