Question 4
Given a rectangular grid, split into $m \times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions:
* all squares touching the border of the grid should be coloured black.
* No four squares forming a $2 \times 2$-square should be coloured in the same colour.
* No four squares forming a $2 \times 2$-square should be coloured in such a way that only the diagonally touching squares have the same colour.
For which grid sizes $m \times n$ (with $m, n \ge 3$) does there exist a valid colouring?
* all squares touching the border of the grid should be coloured black.
* No four squares forming a $2 \times 2$-square should be coloured in the same colour.
* No four squares forming a $2 \times 2$-square should be coloured in such a way that only the diagonally touching squares have the same colour.
For which grid sizes $m \times n$ (with $m, n \ge 3$) does there exist a valid colouring?