RMM 2010 Problem 5

Let $n$ be a given positive integer. Say that a set $K$ of points with integer coordinates in the plane is connected if for every pair of points $R, S\in K$ , there exists a positive integer $\ell$ and a sequence $R=T_0,T_1, T_2,\ldots ,T_{\ell}=S$ of points in $K$ , where each $T_i$ is distance $1$ away from $T_{i+1}$ . For such a set $K$ , we define the set of vectors $$\Delta(K)=\{\overrightarrow{RS}\mid R, S\in K\}$$ What is the maximum value of $|\Delta(K)|$ over all connected sets $K$ of $2n+1$ points with integer coordinates in the plane?

Grigory Chelnokov, Russia