Comment by thedigitalengel

A quick solution involves starting at a corner, and placing each queen one horizontal and two vertical (a constrained knight's move, basically) away from the previous one. This works for an 8x8 board, but does not generalize to an NxN board (it is easy to see that this won't work when N is divisible by 3). Determining which N's it exactly works for is a math, not a programming puzzle, however. :)