3. For a sequence of $2n$ cards numbered $1,2,3, \cdots, 2n$, transform it into $n+1,1, n+2,2, \cdots, n-1,2n, n$, where the original $n$ cards are moved to the $2,4, \cdots, 2n$ positions, and the remaining $n$ cards are placed in the odd positions in their original order. This is called a "perfect" shuffle. Determine for which $n$, starting from the sequence $1,2,3, \cdots, 2n$, the cards can be restored to their original order after a "perfect" shuffle.
3. $n$ can be any positive integer. In fact, let $m=2 n+1$, one shuffle changes $x$ to $y \equiv 2 x(\bmod m)$, so after $k$ shuffles $x$ becomes $2^{k} x(\bmod m)$. Since $m$ is odd, $2^{\phi(m)} \equiv 1(\bmod m)$, thus $2^{\phi(m)} x \equiv x(\bmod m)$, meaning that after $k(=$ $\phi(m))$ shuffles, the original state is restored.