An interesting question came up in the Puzzling Stack Exchange a few days ago about "queen-connected sets". When trying to solve this problem, I came across an arrangement of five colours of queens that would not attack each other on a toroidal 5×5 chessboard, and discovered that 4 queens could be placed into two non-attacking groups but not four individual ones.

Let a *queen number* be a number $n$ where there exists an arrangement of $n$ colours of $n$ queens that are placed on an $n \times n$ chessboard, such that no queen can attack a queen of the same colour, including through broken diagonals.

5 is the smallest queen number, and the arrangement I found is the one where all of them are a knight's move away from each other:

```
1 2 3 4 5
3 4 5 1 2
5 1 2 3 4
2 3 4 5 1
4 5 1 2 3
```

I haven't come across a way to prove that 6 is not a queen number, although I suspect that's true. I also tried making an arrangement for 10 just now using the same method that I used for 5, but it didn't work:

```
1234567890
4567890123
7890123456
0123456789
3456789012
6789012345
9012345678
2345678901
5678901234
8901234567
```

If you notice, every queen attacks the next queen 5 squares right and 5 squares down. But I don't know if there is another arrangement that *does* work.

Is there any way to determine the set of queen numbers without going through every arrangement to verify that it solves the problem?