Repainting of chessboard with restrictions

Question

Assume an $8\times 8$ chessboard with the usual coloring. You may repaint all squares (a) of a row or column (b) of a $2\times 2$ square. The goal is to attain one black square. Can you reach the goal?

I am not asking for the solution because I did not understand the problem properly. Below you see the picture of chessboard with usual coloring. Each row and column has $4$ black and $4$ white squares. If you take any row (or column) and you repaint all squares you still get the row (or column) with $4$ black and $4$ white squares. Any $2\times 2$ square has $2$ black and $2$ white squares and after repainting the number of black and white squares remain unchanged.

Am I missing something? I'd be very thankful if someone can clarify the problem. Thank you so much!

Answer 1

Let $b_i$ and $w_i$ are the number of black and white squares on the chessboard after $i$th step of repainting. Easy to see that $b_i+w_i=64$ with $(b_0,w_0)=(b_1,w_1)=(32,32)$.

Let's consider the following two cases:

i) If we apply repainting of type b) on the $(i+1)$th step, then the following cases are possible: $(b_{i+1},w_{i+1})=(b_{i},w_{i}),\ (b_{i}\pm 2,w_{i}\mp 2),\ (b_{i}\pm 4,w_{i}\mp 4)$ depending on how many black cells $2\times 2$ square has. It follows that $b_{i+1}-w_{i+1}\equiv b_i-w_i \pmod 4$.

ii) If we apply repainting of type a) on the $(i+1)$th step, then we'll get the following situation: WLOG assume that we are repainting column with $x$ black and $y$ white cells. Except that column the chessboard has $b_i-x$ black and $w_i-y$ white cells. Then $(b_i,w_i)\to (b_i-x+y,w_i-y+x)=(b_{i+1},w_{i+1})$. Therefore,
$b_{i+1}-w_{i+1}=(b_i-w_i)-2(x-y)$. Since $x+y=8$, then $2\mid x-y$ and $b_{i+1}-w_{i+1}\equiv b_i-w_i \pmod 4$.

Therefore we were able to show that $b_{i+1}-w_{i+1}\equiv b_i-w_i \pmod 4$.

If it was possible to reach goal, then after $k$th step we would have $(b_k,w_k)=(1,63)$. Since $(b_0,w_0)=(32,32)$ then our claim implies that $b_k-w_k\equiv b_0-w_0\pmod 4$ which is the same as $1-63\equiv 32-32 \pmod 4$ $\Leftrightarrow$ $0\equiv 62\pmod 4$ which is a contradiction.