TITLE: Make all cells of the square be the same color
QUESTION [10 upvotes]: $200 \times 200$ square is colored in chess order. 
In each move we can take any $2 \times 3$  rectangle and change the color of all its cells. 
Can we make all cells of the square be the same color ?

I still do not have an idea. 
Is this construction useful, square with every $2 \times 3$  rectangle contains 2 black cells ?

REPLY [6 votes]: Suppose you managed to make all cells black. Rotate  board so that left-upper corner is black. Now place in each cell with coordinates $(i,j)$ remainder of $(i-j)$ modulo $3$:
$$
\begin{array}{|c|c|c|c|c}
\hline
\bf0 & 1 & \bf2 & 0 &\ldots\\ 
\hline
2 & \bf0 & 1 & \bf2 &\ldots\\
\hline
\bf1 & 2 & \bf0 & 1 &\ldots \\
\hline
0 & \bf1 & 2 &\bf 0 &\ldots\\ 
\hline
\vdots & \vdots & \vdots & \vdots & \ddots
\end{array}
$$
Denote number of black $1$'s as $B$. In beginning $B$ is even, and when all cells are black, $B$ is odd. However, if you repaint all cells in $2\times 3$ rectangle, parity of $B$ remains the same, since there is always two $1$'s in rectangle. That's a contradiction.