13. (25 points) In an $n \times 6 (n \geqslant 2)$ rectangular grid of $6n$ unit squares, each unit square is filled with one of the numbers 0 or 1. If a filling method ensures that there does not exist a rectangular subgrid whose four corner squares are filled with the same number, then it is called an “$N$-filling”, otherwise it is called a “$Y$-filling”. Find the smallest positive integer $n$ such that no matter how the numbers are filled, the filling method is always a $Y$-filling.
13. If $n=4$, the filling method of Table 1 is an $N$-filling method.
Table 1
\begin{tabular}{|l|l|l|l|l|l|}
\hline 1 & 0 & 0 & 1 & 1 & 0 \\
\hline 0 & 1 & 0 & 1 & 0 & 1 \\
\hline 0 & 0 & 1 & 0 & 1 & 1 \\
\hline 1 & 1 & 1 & 0 & 0 & 0 \\
\hline
\end{tabular}

Therefore, $n \geqslant 5$.
Below is the proof: When $n=5$, no matter how the numbers are filled, the filling method is always a $Y$-filling method.
(1) If a certain filling method is a $Y$-filling method, then the filling method remains a $Y$-filling method after arbitrarily swapping any two rows (or two columns).
(2) Consider a $3 \times 6$ rectangular grid.
If a certain filling method is an $N$-filling method, then the filling method of any two columns is not identical. At this point, the filling method of each column in the table cannot have three 1s, nor can it have three 0s. Otherwise, there must be a column that forms a rectangular grid with four identical corner numbers.

Therefore, for a $3 \times 6$ rectangular grid, if the filling method is an $N$-filling method, in terms of the order of rows and columns, the filling method can only be