'''simple docstring'''

from __future__ import annotations







def __UpperCamelCase( _A       :					list[int]	,      _A       :					list[int]	,      _A       :					list[int]	,      _A       :					list[list[str]]	,      _A       :					int	,      ):
   '''simple docstring'''
   UpperCAmelCase__				:					Dict 					=	len(_A     )

   # 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(_A     ):
      # 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]	,      _A	,      _A	,      )







def __UpperCamelCase( _A       :					int     ):
   '''simple docstring'''
   UpperCAmelCase__				:					list[list[str]] 					=	[]
   depth_first_search([]	,      []	,      []	,      _A	,      _A     )

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

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


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  n_queens_solution(4)





'''simple docstring'''

def __UpperCamelCase( _A       :					float	,      _A       :					list[float]     ):
   '''simple docstring'''
   if discount_rate < 0:
      raise ValueError('''Discount rate cannot be negative'''     )
   if not cash_flows:
      raise ValueError('''Cash flows list cannot be empty'''     )
   UpperCAmelCase__				:					str 					=	sum(
       cash_flow / ((1 + discount_rate) ** i) for i, cash_flow in enumerate(_A     )     )
   return round(_A	,      ndigits=2     )


if __name__ == "__main__":
  import doctest

  doctest.testmod()




