2023 USAJMO

Problem 4

Two players,  and , play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with . On ‘s turn,  selects one white unit square and colors it blue. On ‘s turn,  selects two white unit squares and colors them red. The players alternate until  decides to end the game. At this point,  gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score  can guarantee?

(A simple polygon is a polygon (not necessarily convex) that does not intersect itself and has no holes.)