Question 4
A *site* is any point $(x,y)$ in the plane for which $x,y \in \{1, \dots, 20\}$. Initially, each of the $400$ sites is unoccupied. Amy and Ben take turns placing stones on unoccupied sites, with Amy going first; Amy has the additional restriction that no two of her stones may be at a distance equal to $\sqrt5$. They stop once either player cannot move. Find the greatest $K$ such that Amy can ensure that she places at least $K$ stones. Amy places red stones and Ben places blue stones; Ben has no restriction on distances between occupied sites. Amy must guarantee the stated number regardless of Ben's moves.