We are given a $d$-dimensional hypercube $H$, where each vertex is labeled with an integer $\ell\in\{1, 2, \ldots, 2^d\}$. Let $L$ be this labelling.

**Question:** How many labelling permutations $L'$ of $L$ satisfy the property that *each* vertex $v$ of $H$ is labeled in $L'$ with one of the $d$-many labels associated in $L$ with the $d$ vertices adjacent to $v$ (i.e., lying on one of the edges of $v$) in $H$?

**Example:***When $d=2$, the required number of permutations is 4.*

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