Let $S(m)$ denote the sum of the digits of $m$. Among the non-negative integers less than 2000, all multiples of 6 are paired to sum to 1998, forming 167 pairs:
$$
(0,1998),(6,1992),(12,1986), \cdots,
$$
$(996,1002)$.

For each pair of numbers $(x, y)$, let $x=\overline{a_{1} a_{2} a_{3} a_{4}}, y=\overline{b_{1} b_{2} b_{3} b_{4}}$ (if the number has fewer than four digits, pad with 0s at the higher positions). It can be sequentially proven that:
$$
a_{4}+b_{4}=8, a_{3}+b_{3}=9, a_{2}+b_{2}=9, a_{1}+b_{1}=1 \text {. }
$$

Thus, $S(x)+S(y)=27$, indicating that among $S(x)$ and $S(y)$, exactly one is a multiple of 6, meaning that exactly one of $x$ and $y$ is a hexadecagonal number.

Considering that there is exactly one hexadecagonal number, 2004, in the range from 2000 to 2011, the total number of hexadecagonal numbers is $167+1=168$.
3. If a non-negative integer $m$ and the sum of its digits are both multiples of 6, then $m$ is called a "Lucky Six Number". Find the number of Lucky Six Numbers among the non-negative integers less than 2012.
(2012, China Southeast Mathematical Olympiad)