, 2008

Solution:
The cross covers at most five cells so we need at least 4 steps to change the color of every cell. If we place the cross 4 times such that its center lies in the cells marked below, we see that we can turn the whole square black in $n=4$ moves.

Furthermore, applying the same operation twice ("do and undo"), we get that it is possible to turn all the cells black in $n$ steps for every even $n \geq 4$.

We shall prove that for odd $n$ it is not possible to do that. Look at the picture below.

Let $k$ be a difference between white and black cells in the green area in picture. Every figure placed on the table covers an odd number of green cells, so after every step $k$ is changed by a number $\equiv 2 (\bmod 4)$. At the beginning $k=10$, at the end $k=-10$. From this it is clear that we need an even number of steps.

Solution for $n$ is: every even number except $2$.