On an \(8\times 8\) board there is a lamp in every square. Initially every lamp is turned off. In a move we choose a lamp and a direction (it can be the vertical direction or the horizontal one) and change the state of that lamp and all its neighbours in that direction. After a certain number of moves, there is exactly one lamp turned on. Find all the possible positions of that lamp.
In an \(5\times 5\) board one corner was removed. Is it possible to cover the remaining board with linear trominos (\(1\times 3\) blocks)?
Is it possible to cover a \(6 \times 6\) board with the \(L\)-tetraminos without overlapping? The pieces can be flipped and turned.
Is it possible to cover a \((4n+2) \times (4n+2)\) board with the \(L\)-tetraminos without overlapping for any \(n\)? The pieces can be flipped and turned.
Is it possible to cover a \(4n \times 4n\) board with the \(L\)-tetraminos without overlapping for any \(n\)? The pieces can be flipped and turned.
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.