KöMaL 2019 B5046

Let $n\ge3$, and consider the graph in which the vertices are the grid points $(i,j)$, where $1\le i,j\le n$, and the distinct points $(i,j)$ and $(k,l)$ are connected by an edge if and only if $i^2+j^2+k^2+l^2$ is divisible by $3$. For what values of $n$ is it possible to walk the edges of the graph by traversing each edge exactly once?