For any set $X$ set $[X]^2 = \{\{x,y\}: x,y\in X, x\neq y\}$. If $n\geq 2$ is an integer, we endow $\mathbb{Z}$ with a graph structure in the following way. If $x,y\in \mathbb{Z}^n$ we say $x,y$ are *direct neighbors* if there is $k\in\{1,\ldots,n\}$ such that $|x\_k-y\_k|= 1$, and $x\_i = y\_i$ for all $i\in \{1,\ldots,n\}\setminus\{k\}$. Let $$E\_n= \big\{\{x,y\}\in [\mathbb{Z}^n]^2: x,y \text{ are direct neighbors}\}.$$


For which $n\geq 2$ is there an [orientation](https://en.wikipedia.org/wiki/Orientation_(graph_theory)) of $(\mathbb{Z}^n, E\_n)$ such that for every two vertices $x,y\in \mathbb{Z}^n$ there is a directed path from $x$ to $y$?



Orientability of $\mathbb{Z}^n$
Color the vertices of $\mathbb{Z}^n$ by black and white according to their parity. For all but one "direction" (by direction I mean the direction of $x-y$), orient the edges from white to black. Orient the edges of the remaining direction from black to white. Then, for any $x∈\mathbb{Z}^n$ with neighbour $y$, there is a directed path from $x$ to $y$ with length at most three. It follows that, for every two vertices $x,y∈\mathbb{Z}^n$ there is a directed path from $x$ to $y$.


