N-queens (idea)

Heres one neat way to express the N-queen problem. Simple math shows that if two queens can attack each other, one of the following must apply:

• They must be on the same column.
• They must be on the same row.
• The sum of the row and column must be the same for both( \ diagonal).
• The difference of the row and column must be the same for both ( / diagonal).

It can be thought of as a simultaneous equation with 2N unknowns.

Find integers x₁ ... x_n and y₁ ... y_n in the range 1 .. n such that :

Given integers A and B in the range 1..n and A ≠ B

• x_A ≠ x_B ( No two queens on same column )
• y_A ≠ y_B ( No two queens on same row )

For all combinations of integers A and B in the range 1..n and A ≠ B

• (x_A + y_A) ≠ (x_B + y_B) (No two queens on \ diagonal)
• (x_A - y_A) ≠ (x_B - y_B) (No two queens on / diagonal)