Given an $n$ x $n$ board, assume that $n \geq 5$ and that $n$ is not divisible by $2$ or $3$. Prove that the following positioning of $n$ queens $Q_0, Q_2, ..., Q_{n-1}$ works, i.e no two queens threaten each other:
For $0 \leq i \leq n-1$ we position the queen $Q_i$ on the field $(i, 2i \text{ } \pmod n)$.
Here we're using the ($x$-coordinate, $y$-coordinate) coordinate system, where $x$ describes the horizontal position, and $y$ the vertical. For example, in the formula above, $x$ would be $i$, and $y$ would be $2i \pmod n$.
My idea was to prove by contradiction and break up each case on how the queens are positioned to not threaten each other (horizontally, vertically and diagonally), but I can't see what follows. Can someone offer their thoughts or point me in the right direction?