Question

A chessboard is given. It is allowed simultaneously to paint into another colour cells of any vertical or horizontal lines. Is it possible to get a board where there is only one black cell?

Answer 1

When repainting horizontal or vertical lines, consisting of k black and (8-k) white cells, we will get (8-k) black and k white cells. That is why the number of black cells will change to (8-k)-k=8-2k, that is, an 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.