Consider an infinite grid $\mathcal G$ of unit square cells. A chessboard polygon is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$.
Peter chooses a chessboard polygon($F$) and challenges William to paint some cells of $\mathcal G$ green, such that any chessboard polygon congruent to $F$ has at least $1$ green cell but at most $2020$ green cells. Can he choose $F$ to make William's job impossible?