Fix an integer $n \ge 2$. A fairy chess piece [i]leopard [/i] may move one cell up, or one cell to the right, or one cell diagonally down-left. A leopard is placed onto some cell of a $3n \times 3n$ chequer board. The leopard makes several moves, never visiting a cell twice, and comes back to the starting cell. Determine the largest possible number of moves the leopard could have made. 

Dmitry Khramtsov, Russia
To solve this problem, we need to determine the largest possible number of moves a leopard can make on a \(3n \times 3n\) chequerboard, starting and ending at the same cell, without visiting any cell more than once. The leopard can move one cell up, one cell to the right, or one cell diagonally down-left.

1. **Understanding the Moves**:
   - Let \(r\) denote a move to the right.
   - Let \(u\) denote a move up.
   - Let \(d\) denote a move diagonally down-left.

2. **Constraints on Moves**:
   - To return to the starting cell, the number of moves to the right (\(r\)) must equal the number of moves to the left (\(d\)).
   - Similarly, the number of moves up (\(u\)) must equal the number of moves down (\(d\)).
   - Therefore, the total number of moves must be a multiple of 3, i.e., \(3k\) for some integer \(k\).

3. **Maximum Number of Moves**:
   - The total number of cells on the board is \(9n^2\).
   - Since the leopard cannot visit any cell more than once, the maximum number of moves is \(9n^2 - 1\) (returning to the starting cell).

4. **Constructing a Path**:
   - We need to construct a path that uses \(9n^2 - 3\) moves and returns to the starting cell.
   - Consider the following construction for \(n = 2\) and generalize it for larger \(n\):
     - The top 2 rows and the left 2 columns remain unchanged in larger constructions.
     - Stack sub-regions of height three in the remainder of the board.
     - The first row in each sub-region will be almost all right moves.
     - The second row will be all up moves.
     - The third row will be all down-left moves except the leftmost cell.
     - We will get \(n-1\) such regions.
     - From the upper left corner of the uppermost sub-region, move up, right to the edge of the board, and up one square to the top right corner.

5. **Proof of Maximum Moves**:
   - To show that we cannot do better than \(9n^2 - 3\), we use a modulus constraint to restrict the search to tours of length \(9n^2\).
   - No tour can include the top right, top left, and bottom right corners.
   - If we make \(k\) complex moves (\(d\)), we must make exactly \(k\) simple moves of each type (\(r\) and \(u\)) for a total of \(3k\) moves.
   - Therefore, \(9n^2\) is the only possible length of a longer tour.
   - This tour must visit each square on the board, including all 4 corners.
   - Suppose such a tour exists. We may consider the upper right corner to be the start of the tour.
   - We must leave the upper right corner by a down-left move.
   - The tour must include the other 2 corners (bottom right and top left).
   - Consider the path from the upper right square to the bottom right corner.
   - It splits the board into two regions (left and right).
   - It is not possible to move from one region to the other without crossing the path.
   - Therefore, the tour cannot visit all corners, leading to a contradiction.

Thus, the largest possible number of moves the leopard could have made is \(9n^2 - 3\).

The final answer is \(\boxed{9n^2 - 3}\).