The famous N-Queens problem. n_queens(n) returns all the ways in which n queens can be placed on a n x n chessboard so that none of the queens is attacked.

n_queens.py

def n_queens(n):
    def solve():
        i = len(columns)
        if i == n:
            solutions.append([f"{'.' * col}Q{'.' * (n - col - 1)}" for col in columns])
            return
        for j in range(n):
            if is_legal(i, j):
                add_candidate(i, j)
                solve()
                remove_candidate(i, j)

    def is_legal(i, j):
        return not any([j in columns, i - j in major_diagonals, i + j in minor_diagonals])

    def add_candidate(i, j):
        columns.append(j)
        major_diagonals.add(i - j)
        minor_diagonals.add(i + j)

    def remove_candidate(i, j):
        columns.pop()
        major_diagonals.remove(i - j)
        minor_diagonals.remove(i + j)

    columns, major_diagonals, minor_diagonals = [], set(), set()
    solutions = []
    solve()
    return solutions