12. For integers $a_{1}, \ldots, a_{n} \in\{1,2,3, \ldots, 9\}$, we use the notation $\overline{a_{1} a_{2} \ldots a_{n}}$ to denote the number $10^{n-1} a_{1}+10^{n-2} a_{2}+\cdots+10 a_{n-1}+a_{n}$. For example, when $a=2$ and $b=0, \overline{a b}$ denotes the number 20. Given that $\overline{a b}=b^{2}$ and $\overline{a c b c}=(\overline{b a})^{2}$. Find the value of $\overline{a b c}$.
12. Ans: 369.
It is easy to see that $b=5, \overline{a b}=25$ or $b=6, \overline{a b}=36$. Upon checking, $b=6, a=3$ and so $(\overline{b a})^{2}=63^{2}=3969$. Therefore $c=9$. Hence $\overline{a b c}=369$.