Who is a liar?

"Who is a liar?" is puzzle game. The grid is divided into variously sized rectangular "rooms" by orange lines following the edges of the cells. The object of the puzzle is to determine for each cell if it must be painted black or white. The following rules determine which cells are which: Rule 1: Black cells may never be orthogonally connected (they may not share a side, although they can touch diagonally). Rule 2: All white cells must be interconnected (form a single polyomino). Rule 3: A number indicates how many black cells there must be in the around. However, only one number in the each room lies.