Non-attacking queens and symmetry

Place two queens so that neither attacks the other, then three, then more. Two non-attacking queens must avoid sharing a rank, a file, or a diagonal, which already rules out many pairs. Cataloguing solutions is easier with symmetry: the board has eight symmetries (rotations and reflections), so each solution belongs to a family of up to eight, and you can count families instead of individuals.

This is the warm-up for the eight-queens problem. The same constraints scale up, and symmetry keeps the number of essentially different solutions manageable.