Write $1,2,\dots,n^2$ into a $n\times n$ square grid. If $u,v$ are adjacent, call $\max\_{u,v}|u-v|$ the maximal distance of this grid. So what is the minimal maximal distance?

I feel like that it is $n$, the construction is to write $1,2,\dots,n^2$ in order, but I do not know how to prove it, and I cannot find any results on it either.