Question 69
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 exist a positive integer $\ell$ and a sequence $R = T_0, T_1, \dots, T_\ell = S$ 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?
(Russia) Grigory Chelnokov
$$
\Delta(K) = \{ \overrightarrow{RS} \mid R, S \in K \}.
$$
What is the maximum value of $|\Delta(K)|$ over all connected sets $K$ of $2n + 1$ points with integer coordinates in the plane?
(Russia) Grigory Chelnokov