(6) Let $n \geqslant 3$ be a positive integer. If there are $n$ lattice points $P_{1}, P_{2}, \cdots, P_{n}$ in the plane satisfying: when $\left|P_{i} P_{j}\right|$ is a rational number, there exists $P_{k}$ such that $\left|P_{i} P_{k}\right|$ and $\left|P_{j} P_{k}\right|$ are both irrational; when $\left|P_{i} P_{j}\right|$ is an irrational number, there exists $P_{k}$ such that $\left|P_{i} P_{k}\right|$ and $\left|P_{j} P_{k}\right|$ are both rational, then $n$ is called a "good number".
(1) Find the smallest good number;
(2) Is 2005 a good number? (Proposed by Zuming Feng)
We claim that the smallest good number is 5, and 2005 is a good number.
In a triplet $\left(P_{i}, P_{j}, P_{k}\right)$, if $\left|P_{i} P_{j}\right|$ is a rational number (or irrational number), and $\left|P_{i} P_{k}\right|$ and $\left|P_{j} P_{k}\right|$ are irrational numbers (or rational numbers), we call $\left(P_{i}, P_{j}, P_{k}\right)$ a good triplet.
(1) $n=3$ is clearly not a good number.
$n=4$ is also not a good number. If not, assume $P_{1}, P_{2}, P_{3}, P_{4}$ satisfy the condition, and without loss of generality, let $\left|P_{1} P_{2}\right|$ be a rational number and $\left(P_{1}, P_{2}, P_{3}\right)$ be a good triplet, then $\left(P_{2}, P_{3}, P_{4}\right)$ is a good triplet. Clearly, $\left(P_{2}, P_{4}, P_{1}\right)$ and $\left(P_{2}, P_{4}, P_{3}\right)$ are not good triplets. Therefore, $P_{1}, P_{2}, P_{3}, P_{4}$ cannot satisfy the condition. Contradiction!
$n=5$ is a good number. The following five lattice points satisfy the condition:
$$
A_{5}=\{(0,0),(1,0),(5,3),(8,7),(0,7)\} .
$$
(2) Let
$$
\begin{array}{c}
A=\{(1,0),(2,0), \cdots,(669,0)\} . \\
B=\{(1,1),(2,1), \cdots,(668,1)\} . \\
C=\{(1,2),(2,2), \cdots,(668,2)\} . \\
S_{2005}=A \cup B \cup C .
\end{array}
$$

For any positive integer $n$, it is easy to prove that $n^{2}+1$ and $n^{2}+4$ are not perfect squares. It is not difficult to prove that for any two points $P_{i}, P_{j}$ in the set $S_{2005}, \left|P_{i} P_{j}\right|$ is a rational number if and only if $P_{i} P_{j}$ is parallel to one of the coordinate axes. Therefore, 2005 is a good number.
Note When $n=6$,
$$
A_{6}=A_{5} \cup\{(-24,0)\} ;
$$

When $n=7$,
$$
A_{7}=A_{6} \cup\{(-24,7)\} .
$$

Then it can be verified that $n=6$ and 7 are both good numbers.
When $n \geqslant 8$, it can be arranged in three rows like $n=2005$, indicating that for $n \geqslant 8$, all $n$ are good numbers.