Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.

Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$.

[i]Proposed by Carl Schildkraut and Colin Tang[/i]
1. **Define the Problem and Initial Setup:**
   - We have \( n \ge 3 \) players sitting in a circle.
   - Each player draws three cards from a shuffled deck of \( 3n \) cards numbered \( 1, 2, \ldots, 3n \).
   - On each turn, every player passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card one place counterclockwise, while keeping the middle card.
   - Let \( T_r \) denote the configuration after \( r \) turns.

2. **Show that \( T_r \) is Eventually Periodic with Period \( n \):**
   - Define DUE cards as those numbered \( 1, 2, \ldots, n \) and HOANG cards as those numbered \( 2n+1, 2n+2, \ldots, 3n \).
   - DUE cards will always be passed clockwise unless more than one DUE card is held by the same person, in which case the smaller card will be passed clockwise.
   - Similarly, HOANG cards will always be passed counterclockwise unless more than one HOANG card is held by the same person, in which case the larger card will be passed counterclockwise.

3. **Claim: Eventually, Each Person Will Have Exactly One DUE Card:**
   - Let \( 1 \le k \le n \) be the largest integer such that \( 1, 2, \ldots, k \) are all being held by different people.
   - If \( k = n \), we are done.
   - Otherwise, consider the card \( k+1 \), which must be in the hands of someone holding one of the cards \( 1, 2, \ldots, k \).
   - Eventually, cards \( 1, 2, \ldots, k+1 \) will be with different people. This process continues until cards \( 1, 2, \ldots, n \) are with different people.

4. **Similarly, Each Person Will Have Exactly One HOANG Card:**
   - The same argument applies to HOANG cards, ensuring that each person will eventually have exactly one HOANG card.

5. **Periodic Configuration:**
   - At this point, every DUE card will always be passed clockwise, and every HOANG card will always be passed counterclockwise.
   - This ensures that the configuration is periodic with period \( n \).

6. **Determine the Smallest Integer \( m \) for Which \( T_m = T_{m+n} \):**
   - Represent each configuration by labeling each person with the number of DUE cards they have.
   - Let \( a_k \) be the number of DUE cards person \( k \) has, where the players are numbered \( 1, 2, \ldots, n \) in clockwise order.
   - In each turn, the following happens simultaneously for each \( 1 \le k \le n \):
     - If \( a_k = 1, 2 \), then \( a_k \) decreases by 1 and \( a_{k+1} \) increases by 1.
     - If \( a_k = 3 \), then \( a_k \) decreases by 2 and \( a_{k-1}, a_{k+1} \) increase by 1.

7. **Block Representation:**
   - Let a *block* be a group of consecutive nonzero \( a_k \) surrounded by zeroes.
   - After a turn, blocks may merge or remain contiguous.

8. **Claim: Blocks Remain Contiguous:**
   - Zeros cannot appear in the middle of a block and divide it into two or more blocks.

9. **Sum of a Block:**
   - The sum of the numbers within a block is invariant if the block does not merge.

10. **Behavior of Blocks:**
    - If a block does not merge for \( t \) turns, it becomes a \( \min(S, t) \)-ending block, where \( S \) is the sum of the block.

11. **Final Merging:**
    - Eventually, the configuration becomes a single circular block.
    - At turn \( n-1 \), there is an \( n-1 \)-ending block.

12. **Conclusion:**
    - At turn \( n-1 \), each player is guaranteed to have a DUE card and a HOANG card.
    - Therefore, \( m = n-1 \) is the smallest integer for which \( T_m = T_{m+n} \).

The final answer is \( \boxed{ m = n-1 } \).