2. For a regular $n$-sided polygon, construct $n$ squares outside the polygon, each sharing one side with the polygon. It is known that the $2n$ outer vertices of these $n$ squares form a regular $2n$-sided polygon. For what value of $n$ is this possible?
Solve As shown in Figure 1, $P, Q, R$ are three vertices of a regular $n$-sided polygon; $M, N, K, L$ are four vertices of a $2n$-sided polygon. According to the given conditions, we have $MN = NQ, QK = KL, NK = KL = MN$, which means $\triangle NQK$ is an equilateral triangle. Therefore, $\angle NQK = 60^{\circ}$. From the internal angle of the regular $n$-sided polygon $\angle PQR = 120^{\circ}$, we get the exterior angle as $60^{\circ}$.
$$
n \cdot 60^{\circ} = 360^{\circ} \text{, i.e., } n = 6 \text{. }
$$