


'''simple docstring'''



from __future__ import annotations



def      UpperCAmelCase__   (      UpperCAmelCase_       :							list[int] ,       UpperCAmelCase_       :							list[int] ,       UpperCAmelCase_       :							list[int] ,       UpperCAmelCase_       :							list[list[str]] ,       UpperCAmelCase_       :							int ,       )	->						None:
     __lowerCamelCase  :      Tuple														=						len(UpperCAmelCase_				)

     # If row is equal to the size of the board it means there are a queen in each row in
     # the current board (possible_board)
     if row == n:
          # We convert the variable possible_board that looks like this: [1, 3, 0, 2] to
          # this: ['. Q . . ', '. . . Q ', 'Q . . . ', '. . Q . ']
          boards.append(['. ' * i + 'Q ' + '. ' * (n - 1 - i) for i in possible_board]				)
          return

     # We iterate each column in the row to find all possible results in each row
     for col in range(UpperCAmelCase_				):
          # We apply that we learned previously. First we check that in the current board
          # (possible_board) there are not other same value because if there is it means
          # that there are a collision in vertical. Then we apply the two formulas we
          # learned before:
          #
          # 45º: y - x = b or 45: row - col = b
          # 135º: y + x = b or row + col = b.
          #
          # And we verify if the results of this two formulas not exist in their variables
          # respectively.  (diagonal_right_collisions, diagonal_left_collisions)
          #
          # If any or these are True it means there is a collision so we continue to the
          # next value in the for loop.
          if (
              col in possible_board
              or row - col in diagonal_right_collisions
              or row + col in diagonal_left_collisions
          ):
               continue

          # If it is False we call dfs function again and we update the inputs
          depth_first_search(
              [*possible_board, col] ,       [*diagonal_right_collisions, row - col] ,       [*diagonal_left_collisions, row + col] ,       UpperCAmelCase_ ,       UpperCAmelCase_ ,       )



def      UpperCAmelCase__   (      UpperCAmelCase_       :							int				)	->						None:
     __lowerCamelCase  :      list[list[str]]														=						[]
     depth_first_search([] ,       [] ,       [] ,       UpperCAmelCase_ ,       UpperCAmelCase_				)

     # Print all the boards
     for board in boards:
          for column in board:
               print(UpperCAmelCase_				)
          print(''				)

     print(len(UpperCAmelCase_				) ,       'solutions were found.'				)


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      n_queens_solution(4)










'''simple docstring'''



def      UpperCAmelCase__   (      UpperCAmelCase_       :							int = 1_00				)	->						int:
     __lowerCamelCase  :      Union[str, Any]														=						n * (n + 1) * (2 * n + 1) / 6
     __lowerCamelCase  :      Union[str, Any]														=						(n * (n + 1) / 2) ** 2
     return int(square_of_sum - sum_of_squares				)


if __name__ == "__main__":
      print(f'''{solution() = }''')






