The eight queens puzzle

Place eight queens so that no two attack each other: none sharing a rank, file, or diagonal. Many solutions exist if you count every distinct placement; far fewer remain once rotations and reflections of the board are treated as the same solution. A good way in is Polya's partial-solution method: start from a valid placement of a few queens and extend it, backing up whenever a queen has no safe square.

The constraints are strict enough that most partial placements fail, which is why systematic search (place, check, backtrack) beats random trial. Symmetry is a bookkeeping tool: it groups congruent solutions so you do not recount mirrors of the same arrangement.