11. (16 points) For an integer $k$, define the set
$$
S_{k}=\{n \mid 50 k \leqslant n<50(k+1), n \in \mathbf{Z}\} \text {. }
$$

How many of the 600 sets $S_{0}, S_{1}, \cdots, S_{599}$ do not contain any perfect squares?
11. Notice,
$$
\begin{array}{l}
(x+1)^{2}-x^{2}=2 x+1 \leqslant 50(x \in \mathbf{N}) \\
\Leftrightarrow x \leqslant 24(x \in \mathbf{N}) .
\end{array}
$$

While $(24+1)^{2}=625 \in S_{12}$, thus, the square numbers in $S_{0}, S_{1}$, $\cdots, S_{12}$ do not exceed $25^{2}$, and each set consists of 50 consecutive non-negative integers. Therefore, each set contains at least 1 square number.

In the sets $S_{13}, S_{14}, \cdots, S_{599}$, if they contain square numbers, they are no less than $26^{2}$.

When $x \geqslant 26$, $2 x+1 \geqslant 53$, thus, in $S_{13}, S_{14}$, $\cdots, S_{599}$, each set contains at most 1 square number.
On the other hand, the largest number in $S_{599}$ is
$$
600 \times 50-1=29999 \text {, }
$$

and $173^{2}<29999<174^{2}$, so the square numbers in $S_{13}, S_{14}, \cdots, S_{599}$ do not exceed $173^{2}$.

Therefore, there are exactly $173-$ $25=148$ sets in $S_{13}, S_{14}, \cdots, S_{599}$ that contain square numbers.
In summary, in $S_{0}, S_{1}, \cdots, S_{599}$, there are
$$
600-13-148=439
$$

sets that do not contain square numbers.