In ZS Chess, an Ivanight attacks like a knight, except that if the attacked square is out of range, it goes
through the edge and comes out from the other side of the board, and attacks that square instead. The
ZS chessboard is an $8 \times 8$ board, where cells are coloured with $n$ distinct colours, where $n$ is a natural
number, such that a Ivanight placed on any square attacks $ 8 $ squares that consist of all $n$ colours, and
the colours appear equally many times in those $ 8 $ squares. For which values of $n$ does such a ZS chess
board exist?
To determine for which values of \( n \) such a ZS chessboard exists, we need to ensure that an Ivanight placed on any square attacks 8 squares that consist of all \( n \) colors, and the colors appear equally many times in those 8 squares.

1. **Understanding the Ivanight's Movement:**
   An Ivanight moves like a knight in standard chess, which means it can move to any of the 8 possible positions: \((\pm 2, \pm 1)\) and \((\pm 1, \pm 2)\). However, if the move takes it out of the board, it wraps around to the other side.

2. **Color Distribution:**
   We need to color the \( 8 \times 8 \) board such that each Ivanight's 8 possible attack squares contain all \( n \) colors equally. This implies that \( n \) must divide 8, as each color must appear exactly once in the 8 attack squares.

3. **Possible Values of \( n \):**
   The divisors of 8 are 1, 2, 4, and 8. We need to check if these values of \( n \) can satisfy the condition.

4. **Case \( n = 1 \):**
   If \( n = 1 \), the entire board is of one color. This trivially satisfies the condition since all 8 attacked squares will be of the same color.

5. **Case \( n = 2 \):**
   If \( n = 2 \), we need to color the board such that each Ivanight's 8 attack squares contain both colors equally. This can be achieved by alternating colors in a checkerboard pattern.

6. **Case \( n = 4 \):**
   If \( n = 4 \), we need to color the board such that each Ivanight's 8 attack squares contain all 4 colors equally. This can be achieved by repeating a \( 2 \times 2 \) block pattern of the 4 colors.

7. **Case \( n = 8 \):**
   If \( n = 8 \), we need to color the board such that each Ivanight's 8 attack squares contain all 8 colors equally. This can be achieved by repeating a \( 4 \times 2 \) or \( 2 \times 4 \) block pattern of the 8 colors.

   The provided solution shows a possible configuration for \( n = 8 \):

   \[
   \begin{array}{llllllll}
   1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
   3 & 4 & 5 & 6 & 7 & 8 & 1 & 2 \\
   3 & 4 & 5 & 6 & 7 & 8 & 1 & 2 \\
   1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
   5 & 6 & 7 & 8 & 1 & 2 & 3 & 4 \\
   7 & 8 & 1 & 2 & 3 & 4 & 5 & 6 \\
   7 & 8 & 1 & 2 & 3 & 4 & 5 & 6 \\
   5 & 6 & 7 & 8 & 1 & 2 & 3 & 4
   \end{array}
   \]

   This configuration ensures that each Ivanight's 8 attack squares contain all 8 colors equally.

Conclusion:
The values of \( n \) for which such a ZS chessboard exists are \( n = 1, 2, 4, 8 \).

The final answer is \( \boxed{ n = 1, 2, 4, 8 } \).