Suppose we have a pack of $2n$ cards, in the order $1, 2, . . . , 2n$. A perfect shuffle of these cards changes the order to $n+1, 1, n+2, 2, . . ., n- 1, 2n, n$ ; i.e., the cards originally in the first $n$ positions have been moved to the places $2, 4, . . . , 2n$, while the remaining $n$ cards, in their original order, fill the odd positions $1, 3, . . . , 2n - 1.$
Suppose we start with the cards in the above order $1, 2, . . . , 2n$ and then successively apply perfect shuffles. 
What conditions on the number $n$ are necessary for the cards eventually to return to their original order? Justify your answer.

[hide="Remark"]
Remark. This problem is trivial. Alternatively, it may be required to find the least number of shuffles after which the cards will return to the original order.[/hide]
1. **Understanding the Perfect Shuffle:**
   - We start with a deck of $2n$ cards in the order $1, 2, \ldots, 2n$.
   - A perfect shuffle rearranges the cards such that the new order is $n+1, 1, n+2, 2, \ldots, 2n, n$.
   - Specifically, the cards originally in the first $n$ positions move to the even positions $2, 4, \ldots, 2n$, and the remaining $n$ cards fill the odd positions $1, 3, \ldots, 2n-1$.

2. **Position Transformation:**
   - Let $k$ be the original position of a card. After one perfect shuffle, the card in position $k$ moves to position $2k \pmod{2n-1}$.
   - This is because the new position is determined by doubling the original position and taking modulo $2n-1$ to ensure it wraps around correctly.

3. **Generalizing the Transformation:**
   - After $r$ perfect shuffles, the card originally in position $k$ will be in position $2^r k \pmod{2n-1}$.
   - We need to determine when the deck returns to its original order, which happens when $2^r k \equiv k \pmod{2n-1}$ for all $k$.
   - This simplifies to $2^r \equiv 1 \pmod{2n-1}$.

4. **Order of 2 Modulo $2n-1$:**
   - The problem reduces to finding the smallest positive integer $r$ such that $2^r \equiv 1 \pmod{2n-1}$.
   - This $r$ is known as the order of 2 modulo $2n-1$, denoted as $\text{ord}_{2n-1}(2)$.

5. **Existence of $r$:**
   - By Euler's theorem, since $2$ and $2n-1$ are coprime, there exists an integer $r$ such that $2^r \equiv 1 \pmod{2n-1}$.
   - The Well-Ordering Principle guarantees that there is a smallest positive integer $r$ for which this congruence holds.

6. **Conclusion:**
   - There are no specific restrictions on $n$ other than $n \in \mathbb{Z}$ and $n > 0$.
   - For any $n$, the cards will eventually return to their original order after a certain number of perfect shuffles, determined by the order of 2 modulo $2n-1$.

\[
\boxed{n \in \mathbb{Z} \mid n > 0}
\]