Question

A chessboard is given. It is allowed simultaneously to paint into another colour all cells located in a square 2x2. Is it possible to get only one black cell on the board?

Answer 1

When repainting a square with sizes 2x2, consisting of k black and (4-k) white cells, we will get (4-k) black and k white cells. That is why the number of black cells will change to (4-k)-k= 4-2k, that is, even number. Taking into account a fact that evenness of number of black cells does not change, from initial 32 black cells we won’t be able to get one black cell.