Example 6 If there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1,2, \cdots, n$, such that $k+a_{k} (k=1,2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11,13,15,17,19\}$, which are "good numbers" and which are not "good numbers"? Explain your reasoning!
(2004 China Girls Mathematical Olympiad)
Analysis: Since $\{11,13,15,17,19\}$ is a finite set and contains only 5 elements, we can discuss these elements one by one.

Solution (1) 11 is not a "good number," because 4 can only be added to 5 to get $3^{2}$, and 11 can only be added to 5 to get $4^{2}$, thus there does not exist a permutation that meets the requirement.
(2) 13 is a "good number," because in the following permutation, $k+a_{k}(k=1,2, \cdots, 13)$ are all perfect squares.
$$
\begin{array}{llcccccccccccc}
k: & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\
a_{k}: & 8 & 2 & 13 & 12 & 11 & 10 & 9 & 1 & 7 & 6 & 5 & 4 & 3
\end{array}
$$
(3) 15 is a "good number," because in the following permutation, $k+a_{k}(k=1,2, \cdots, 15)$ are all perfect squares.
$$
\begin{array}{lccccccccccccccc}
k: & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\
a_{k}: & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1
\end{array}
$$
(4) 17 is a "good number," because in the following permutation, $k+a_{k}(k=1,2, \cdots, 17)$ are all perfect squares.
$$
\begin{array}{llllllcccccccccccc}
k: & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\
a_{k}: & 3 & 7 & 6 & 5 & 4 & 10 & 2 & 17 & 16 & 15 & 14 & 13 & 12 & 11 & 1 & 9 & 8
\end{array}
$$
(5) 19 is a "good number," because in the following permutation, $k+a_{k}(k=1,2, \cdots, 19)$ are all perfect squares.
$$
\begin{array}{llllllllllllllllllll}
k: & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 \\
a_{k}: & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 16 & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 19 & 18 & 17
\end{array}
$$

In summary, in the set $\{11,13,15,17,19\}$, all elements except 11 are "good numbers".