Question 23
Consider a $2 \times 3$ grid where each entry is one of $0$, $1$, and $2$. For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$? One valid grid is shown below.
$$
\left[\begin{array}{lll}
1 & 2 & 0 \\
2 & 1 & 0
\end{array}\right]
$$
$$
\left[\begin{array}{lll}
1 & 2 & 0 \\
2 & 1 & 0
\end{array}\right]
$$