12. Using 6 different colors to color $\mathrm{n}(\mathrm{n} \geq 2)$ interconnected regions $\mathrm{A}_{1}, \mathrm{~A}_{2}, \ldots, \mathrm{A}_{\mathrm{n}}$, such that any two adjacent regions are colored differently, the number of all different coloring schemes $a_{\mathrm{n}}=$ $\qquad$
Solution: Let the number of coloring schemes that meet the requirements be denoted as $a_{\mathrm{n}}$. Then, region $\mathrm{A}_{1}$ has 6
coloring methods, and regions $\mathrm{A}_{2}, \mathrm{~A}_{3}, \ldots, \mathrm{A}_{\mathrm{n}}$ each have 5 coloring methods, which include the cases where $\mathrm{A}_{\mathrm{n}}$ and $\mathrm{A}_{1}$ are colored the same or differently: if region $\mathrm{A}_{\mathrm{n}}$ and $\mathrm{A}_{1}$ are the same color, then consider $\mathrm{A}_{\mathrm{n}}, \mathrm{A}_{1}$ as one
region, making a total of $\mathrm{n}-1$ regions, with $a_{\mathrm{n}-1}$ coloring methods that meet the requirements; if region $\mathrm{A}_{\mathrm{n}}$ and $\mathrm{A}_{1}$ are colored differently, then there are
$a_{\mathrm{n}}$
coloring methods. Therefore, we have $a_{\mathrm{n}}+a_{\mathrm{n}-1}=6 \times 5^{\mathrm{n}-1}$, which means $\left(a_{\mathrm{n}}-5^{\mathrm{n}}\right)=-\left(a_{\mathrm{n}-1}-5^{\mathrm{n}-1}\right)$. Since $a_{2}=6 \times 5=30, a_{2}-5^{2}=5$, we have $a_{\mathrm{n}}-5^{\mathrm{n}}=5(-1)^{\mathrm{n}-2}=5(-1)^{\mathrm{n}}$, hence $a_{\mathrm{n}}=5(-1)^{\mathrm{n}}+5^{\mathrm{n}}$.