Question 149
The unit squares of an $N \times N$ board are coloured black and white so that squares that share a side have different colours, and so that at least one corner square is coloured black. In each step we choose a $2 \times 2$ square and change the colour of all four unit squares inside that square, so that white unit squares become black, black become grey and grey become white.
Determine all positive integers $N \gt 1$ for which it is possible, using a finite number of steps, to achieve that all unit squares that were originally black become white, and all unit squares that were originally white become black.
(Russia 2012)
Determine all positive integers $N \gt 1$ for which it is possible, using a finite number of steps, to achieve that all unit squares that were originally black become white, and all unit squares that were originally white become black.
(Russia 2012)