Mochinyoro

Paint some empty cells black. No square of side 2 is entirely black. No black area is rectangular. All white areas are rectangular. All white areas are diagonally adjacent. A white area may contain up to one number, and that number must be its area.

The following puzzle is intended to be instructive, so not terribly hard.

http://pzv.jp/p.html?mochinyoro/10/10/q-14l6zzzhcleq