Stostone Rules

Stostone is a new puzzle type by Nikoli. Shade exactly one stone, a connected group of cells, in each region, such that no two stones in different regions are adjacent.┬áIf a number is in a region, it tells the size of that region’s stone. If you let the stones all fall down, they must cover exactly the bottom half of the grid.

See the example below, where the 1st grid is the puzzle, the 2nd grid is the solution, and the 3rd grid is what it looks like if all the stones fall down.