18. Given a positive integer $n \geqslant 2$. On an infinite grid paper, $n$ cells are colored red. A rectangular grid array is called "special" if and only if it contains at least two red cells on its diagonals. A single red cell is also considered special. A row or column grid arrangement is also special if both ends are red cells. Given a fixed pattern of $n$ red cells, let $N$ denote the maximum number of red cells that a special grid array can contain. Determine the minimum value of $N$ among all possible patterns of $n$ red cells.
18. The minimum value sought is $1+\left\lceil\frac{n+1}{5}\right\rceil$.
(1) Given $n$ fixed red cells, we need to prove that $N \geqslant \frac{n+6}{5}$.

Consider the smallest rectangular grid array $A$ containing the given $n$ red cells.

By minimality, the bottommost row, the rightmost column, the topmost row, and the leftmost column of $A$ each contain red cells $a, b, c, d$ (not necessarily distinct); for example, $a$ is the cell at the bottom-left corner, and the list can be written as $a, b, c, a$.

Let $[x y]$ denote the (unique) rectangular grid array with diagonal cells $x$ and $y$.

Notice that the special rectangular grid arrays $[a b], [b c], [c d], [d a]$, and $[a c]$ can cover $A$, where the first two cover the right side of $[a c]$, and the last two cover the left side of $[a c]$. Considering repeated counting, $a$ and $c$ are contained in three of these special rectangular grid arrays, $b$ and $d$ are contained in two of these special rectangular grid arrays, and the other red cells are contained in at least one of these special rectangular grid arrays.
Let $r_{[x y]}$ denote the number of red cells contained in $[x y]$, then
\[ r_{[a b]} + r_{[b c]} + r_{[c d]} + r_{[d a]} + r_{[a c]} \geqslant 3 \times 2 + 2 \times 2 + (n-4) = n + 6 \]
\[ \Rightarrow N \geqslant \frac{n+6}{5} \]
(2) Provide a construction of $n$ red cells such that
\[ N = 1 + \left\lceil\frac{n+1}{5}\right\rceil \]
Let $m = \left\lceil\frac{n+1}{5}\right\rceil$, i.e.,
\[ n = 5m - r \quad (1 \leqslant r \leqslant 5, r \in \mathbf{Z}) \]
and $N = m + 1$.
Fix an integer $k > 2m$, and let $S$ be a $3k \times 3k$ square grid, divided into 9 $k \times k$ sub-square grids.

Let $S_{L}$ denote the $k \times k$ sub-square grid at the bottom-left corner, and sequentially color $m$ cells on the diagonal starting from the bottom-right corner red.

In the following discussion, the maximum and minimum values are only valid for the first few cases where $m < r$. If $n \geqslant 20$, then clearly $m \geqslant r$, and the maximum and minimum values do not need to be considered.

Let $S_{U L}$ denote the $k \times k$ sub-square grid at the top-left corner, and sequentially color $\min \{m, 4m - r\}$ cells on the diagonal starting from the bottom-left corner red.

Let $S_{U R}$ denote the $k \times k$ sub-square grid at the top-right corner, and sequentially color $\min \{m, 3m - r\}$ cells on the diagonal starting from the top-left corner red.

Let $S_{L R}$ denote the $k \times k$ sub-square grid at the bottom-right corner, and sequentially color $\min \{m, 2m - r\}$ cells on the diagonal starting from the top-right corner red.

Let $S_{C}$ denote the $k \times k$ sub-square grid in the center, and color $\max \{0, m - r\}$ cells in $S_{C}$ red.
At this point, exactly $n$ cells are colored red.
Notice that for any two "adjacent" $k \times k$ sub-square grids: $S_{L L}$ and $S_{U L} \backslash S_{U L}$ and $S_{U R} \backslash S_{U R}$ and $S_{L R}$, $S_{L R}$ and $S_{L L}$, there are both horizontal and vertical grid lines separating the red cells they contain.

To complete the argument, we need to prove that if $x \backslash y$ are two red cells in this construction, then
\[ r_{[x y]} \leqslant m + 1 \]
If $x$ and $y$ are both in one of $S_{L L}, S_{U L}, S_{U R}, S_{L R}, S_{C}$, then the above inequality clearly holds, because each of the five $k \times k$ sub-square grids contains at most $m$ red cells.

If $x$ and $y$ are in two of the adjacent $k \times k$ sub-square grids mentioned earlier, then $[x y]$ contains red cells from these two sub-square grids. Additionally, $[x y]$ contains only one red cell in one of the sub-square grids, which is either $x$ or $y$ itself. Thus,
\[ r_{[x y]} \leqslant m + 1 \]
If $x$ and $y$ are in diagonal $k \times k$ sub-square grids, i.e., $S_{U L}$ and $S_{U R} \backslash S_{U L}$ and $S_{L R}$, for fixed $x$ and $y$, clearly, $[x y]$ does not contain red cells from the other diagonal $k \times k$ sub-square grids, and in the diagonal $k \times k$ sub-square grids it belongs to, it only contains the two red cells $x \backslash y$. The remaining red cells it contains are all in $S_{C}$, thus,
\[ r_{[x y]} \leqslant 2 + (m - 1) = m + 1 \]
Finally, if one of $x$ and $y$ is in $S_{C}$ and the other is in one of $S_{L L} \backslash S_{U L} S_{U R} S_{L R}$, say $S'$, then $[x y]$ contains only one red cell in $S'$. Thus,
\[ r_{[x y]} \leqslant 1 + (m - 1) = m < m + 1 \]