27.1.9 $\star \star$ 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".

Determine which numbers in the set $\{11,13,15,17,19\}$ are "good numbers" and which are not. Provide reasons for your answer.
Parse: Except for 11, all are “good numbers”.
(1) It is not difficult to know that 11 can only be added to 5 to get $4^{2}$, and 4 can only be added to 5 to get $3^{2}$, so there is no arrangement that satisfies the condition, hence 11 is not a “good number”.
(2) 13 is a “good number”, because in the following table, $k+a_{k}(k=1,2,3, \cdots, 13)$ are all perfect squares:
$$
\begin{array}{lllllllllllllll}
k: & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\
a_{k}: & 8 & 2 & 13 & 12 & 11 & 10 & 9 & 1 & 7 & 6 & 5 & 4 & 3
\end{array}
$$
(3) 15 is a “good number”, because in the following table, $k+a_{k}(k=1,2, \cdots, 15)$ 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 table, $k+a_{k}(k=1,2, \cdots, 17)$ are all perfect squares:
$$
\begin{array}{lrrrrrcccccccccccc}
k: & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\
a_{k}: & 3 & 7 & 6 & 5 & 4 & 10 & 2 & 17 & 16 & 15 & 14 & 13 & 12 & 11 & 1 & 9 & 8
\end{array}
$$

Here, the cycle $(1,3,6,10,15,1)$ is used.
(5) 19 is a “good number”, because in the following table, $k+a_{k}(k=1,2, \cdots, 19)$ are all perfect squares:
$$
\begin{array}{llllllllllllllllllll}
k: & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 \\
a_{k}: & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 16 & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 19 & 18 & 17
\end{array}
$$