Can you cover a \(13 \times 13\) square using \(2 \times 2\) and \(3 \times 3\) squares?
A \((2n - 1) \times (2n - 1)\) board is tiled with pieces of the following possible types:
Prove that at least \(4n-1\) of the first type have been used.
In the \(n\times n\) table, the two opposite corner squares are black and the rest are white. Find the smallest number of white cells that is enough to be repainted black in order to make all the cells of the table black with only there transformations: repaint all the cells of one column, or all the cells of one row into the opposite colour.