Queens problem is solvable
If I understand the problem ... you have an N by N matrix where you need to find the set of solutions where you can only have a piece in a square that is neither in the same row or column of an existing piece, nor is on a diagonal to an existing piece.
I think this is the solution, which I tested on the 4x4 and 5x5 matrices.
There is a distinct pattern...
The matrix is numbered 0 thru N-1;
Lower left is 0,0 while upper right is (N-1, N-1)
You move from left to right when placing queens.
Place the first piece in column 0 in any square. (0,k)
Block out the row;
Block out the column;
Block out the diagonal(s) to the right;
In the next column you place the queen in the square MOD(k+j, N) where j in {2 thru (N-2)}
If the square is blocked, then the solution attempt fails and you move on to the next possible choice.
And so on. ...
This works for N=4 and N=5
I did it by hand, but will now cobble up a simple program to do it and for larger sets.
Of course then you have to run thru all of the possibilities to show that this is wrong. But I don't think it is.
Now where do I submit my solution?
Note: If the order of queens matters, then its merely an N! for each solution within the set.