Selecting cells so that every $2\times 2$ square is odd, then even Jacob selects some cells from a $12\times9$ table, so that every $2\times 2$ subsquare contains an odd number of selected cells. He then selects some more cells, so that every $2\times 2$ subsquare now contains an even number of selected cells. What is the minimum number of selected cells at the end?
[Source: Russian competition problem]
 A: If, at the end, every $2\times2$ subsquare now contains an even number of selected cells, then every $2\times2$ subsquare now contains either $2$ or $4$ cells. (A $2\times2$ subsquare cannot have $0$ selected cells, since at the first stage it had at least $1$ selected cell.)
Considering the $24$ disjoint $2\times2$ subsquares that make up the leftmost $12\times8$ section of the table, each has at least $2$ selected cells in the end. So these account for at least $48$ selected cells. So $48$ is a lower bound for the answer. But we can actually carry out the selection in a way that selects $48$ cells.
Consider if these $48$ selected cells were the contents of every other column, starting from the second. Is this compatible with us having gone through the first stage, where every $2\times2$ subsquare contained an odd number of selected cells? Sure, just unselect from every other row. 
So $48$ is the minimal number of selected cells at the end. Here is one selection process for which this works.
$$\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline\end{array}
$$
$$\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}&\phantom{\checkmark}\\
\hline\end{array}
$$
$$\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline
\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}&\checkmark&\phantom{\checkmark}\\
\hline\end{array}
$$
