7. On a square table composed of unit squares of size $2011 \times 2011$, a finite number of napkins are placed, each covering a $52 \times 52$ square. In each unit square, write the number of napkins covering it, and let the maximum number of unit squares with the same number be $k$. For all possible configurations of napkins, find the maximum value of $k$.


7. The maximum value of $k$ is
$2011^{2}-\left[\left(52^{2}-35^{2}\right) \times 39-17^{2}\right]$
$=4044121-57392=3986729$.
Let $m=39$. Then $2011=52 m-17$.
Below is an example where there are 3986729 unit squares with the same number written in them.
Let the column numbers from left to right, and the row numbers from bottom to top be
$1,2, \cdots, 2011$.
Each napkin is represented by the coordinates of its lower left corner, and the napkins are divided into four categories:
The coordinates of the first category are
$(52 i+36,52 j+1)(0 \leqslant j \leqslant i \leqslant m-2)$;
The coordinates of the second category are
$(52 i+1,52 j+36)(0 \leqslant i \leqslant j \leqslant m-2)$;
The coordinates of the third category are
$(52 i+36,52 i+36)(0 \leqslant i \leqslant m-2)$;
The coordinates of the fourth category are $(1,1)$.
In Figure 2, different shading indicates different categories of napkins.
Except for the unit squares with at least two different types of shading, the number written in all other unit squares is 1. It is easy to calculate that the number of unit squares with at least two different types of shading is
$\left(52^{2}-35^{2}\right) m-17^{2}=57392$.
Below is the proof: the number of unit squares with the same number written in them does not exceed 3986729.

For any configuration of napkins and any positive integer $M$, assume there are $g$ unit squares with numbers different from $M$. We need to prove that $g \geqslant 57392$.
A line is defined as either a row or a column.
Consider any line $l$, and let the numbers written in the adjacent unit squares be $a_{1}, a_{2}, \cdots, a_{52 m-17}$.

For $i=1,2, \cdots, 52$, let $s_{i}=\sum_{t=i(\bmod 52)} a_{t}$, where $s_{1}, s_{2}, \cdots, s_{35}$ each have $m$ terms, and $s_{36}, s_{37}, \cdots, s_{52}$ each have $m-1$ terms. Each napkin intersecting line $l$ contributes exactly 1 to each $s_{i}$.
Thus, the number of napkins intersecting $l$, denoted as $s$, satisfies
$s_{1}=s_{2}=\cdots=s_{52}=s$.
If $s>(m-1) M$, then line $l$ is called "rich"; otherwise, it is called "poor".

Assume $l$ is rich. Then each of $s_{36}, s_{37}, \cdots, s_{52}$ has at least one term greater than $M$. Consider all the unit squares corresponding to these terms, and call them "rich bad squares" for this line. Thus, each rich line has at least 17 unit squares that are rich bad squares for this line.
On the other hand, if $l$ is poor, then
$s \leqslant(m-1) M<m M$.
Thus, each of $s_{1}, s_{2}, \cdots, s_{35}$ has at least one term less than $M$.

Consider all the unit squares corresponding to these terms, and call them "poor bad squares" for this line. Thus, each poor line has at least 35 unit squares that are poor bad squares for this line.

Call the labels "small" if they are congruent to $1,2, \cdots, 35$ modulo 52, and "large" otherwise, i.e., congruent to $36,37, \cdots, 52$ modulo 52.

Considering the column numbers from left to right and the row numbers from bottom to top are $1,2, \cdots, 52 m-17$, a line is called "large" or "small" depending on whether its label is large or small.

By definition, the rich bad squares of a row belong to large columns, and the poor bad squares of a row belong to small columns, and vice versa.

On each line, place a strawberry in each bad square for that line, and additionally, place an extra strawberry in each rich bad square for each "small rich" line. A unit square can independently receive strawberries from its row and column.

Note that a unit square with one strawberry has a number different from $M$. If this unit square receives an extra strawberry, then the number in this unit square is greater than $M$. Thus, it is either in a small row or a large column, and vice versa. Assume it is in a small row, then it is not bad for this column. Thus, in this case, it has no more than two strawberries. If the extra rule does not apply to this unit square, then it also has no more than two strawberries. Thus, the total number of strawberries $N$ is at most $2 g$.
Below, we estimate $N$ using a different method.
For the $2 \times 35 \mathrm{~m}$ small lines, if it is rich, there are at least 34 strawberries; if it is poor, there are at least 35 strawberries. Therefore, in either case, there are at least 34 strawberries.

Similarly, for the $2 \times 17(m-1)$ large lines, there are at least $\min \{17,35\}=17$ strawberries.
Summing over all lines, we get
$$
\begin{array}{l}
2 g \geqslant N \geqslant 2[35 m \times 34+17(m-1) \times 17] \\
=2(1479 m-289)=2 \times 57392, \\
g \geqslant 57392 .
\end{array}
$$

Thus, $g \geqslant 57392$.