Elementary proof techniques for the maximum number of islands

Islands are combinatorial objects that can be intuitively defined on a board consisting of a finite number of cells. It is a fundamental property that two islands are either containing or disjoint. Czédli determined the maximum number of rectangular islands. Pluhár solved the same problem for bricks, and Horváth, Németh and Pluhár for triangular islands. Here, we give a much shorter proof for these results, and also for new, analogous results on toroidal and some other boards.