Question 161
In the following squares, we are allowed to replace all the 0s by 1s and vice versa on an entire row, an entire column or an entire diagonal. In each case, can we obtain only 0s?
$$
\left(\begin{array}{llll}
0 & 1 & 1 & 0 \\
1 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1
\end{array}\right) \quad\left(\begin{array}{llll}
0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 \\
1 & 0 & 1 & 1
\end{array}\right) \quad\left(\begin{array}{llll}
0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & 0 & 1
\end{array}\right)
$$
$$
\left(\begin{array}{llll}
0 & 1 & 1 & 0 \\
1 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1
\end{array}\right) \quad\left(\begin{array}{llll}
0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 \\
1 & 0 & 1 & 1
\end{array}\right) \quad\left(\begin{array}{llll}
0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & 0 & 1
\end{array}\right)
$$