KöMaL 2023 B5310

A game of strategy is played by four teams on a map represented by an $n \times n$ square grid ($n \ge 3$). Every square field is either sea or land. The bases of the four teams are in the four corners of the map, on land. Given that there is one large connected region of sea on the map and no two bases are connected by a path on land, determine the minimum possible number of fields that represent the sea. (A path consists of adjacent fields. Two fields are adjacent if they have an edge in common. The region of sea is connected in this sense: any two sea squares can be connected by a path consisting of sea squares.)