Let $G=(V,E)$ be an undirected graph, such that $V$ is a subset of $\{1,\dotsc,N\} \times \{1,\dotsc,N\}$ and there is an edge between $2$ vertices if and only if they have one identical coordinate and differ by exactly $1$ in the other coordinate (we can think about it as a grid, with some points omitted, such that we can move up/down/right/left).
The graph $G$ defines a metric (by $d(v_1,v_2)=$minimum number of edges in a path from $v_1$ to $v_2$). A sphere is defined as usual: for $v \in V$, we let $S(v,r)=\{u \in V \mid d(v,u)=r\}$.
What is $\max\limits_{v,r}|S(v,r)|$? An asymptotic expression in terms of $N$ would be enough. I strongly suspect it is $O(N)$ because that is what we get if $V=\{1,\dotsc,N\} \times \{1,\dotsc,N\}$ (i.e: no point omitted) and take $v=(N/2,N/2)$ and $r=N/2$.