


from __future__ import annotations



def 						lowerCAmelCase		(		_lowerCAmelCase							:   list[int]						,							_lowerCAmelCase							:   list[int]						,							_lowerCAmelCase							:   list[int]						,							_lowerCAmelCase							:   list[list[str]]						,							_lowerCAmelCase							:   int						,							):




 """simple docstring"""







 UpperCAmelCase__							 =						len(_lowerCAmelCase	)

 # 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(_lowerCAmelCase	):
  # 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]						,							_lowerCAmelCase						,							_lowerCAmelCase						,							)



def 						lowerCAmelCase		(		_lowerCAmelCase							:   int	):




 """simple docstring"""







 UpperCAmelCase__							 =						[]
 depth_first_search([]						,							[]						,							[]						,							_lowerCAmelCase						,							_lowerCAmelCase	)

 # Print all the boards
 for board in boards:
  for column in board:
   print(_lowerCAmelCase	)
  print(""	)

 print(len(_lowerCAmelCase	)						,							"solutions were found."	)


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  n_queens_solution(4)











import math



def 						lowerCAmelCase		(		_lowerCAmelCase							:   int	):




 """simple docstring"""







 UpperCAmelCase__							 =						math.loga(math.sqrt(4 * positive_integer + 1	) / 2 + 1 / 2	)

 return exponent == int(_lowerCAmelCase	)



def 						lowerCAmelCase		(		_lowerCAmelCase							:   float = 1 / 1_2345	):




 """simple docstring"""







 UpperCAmelCase__							 =						0
 UpperCAmelCase__							 =						0

 UpperCAmelCase__							 =						3
 while True:
  UpperCAmelCase__							 =						(integer**2 - 1) / 4
  # if candidate is an integer, then there is a partition for k
  if partition_candidate == int(_lowerCAmelCase	):
   UpperCAmelCase__							 =						int(_lowerCAmelCase	)
   total_partitions += 1
   if check_partition_perfect(_lowerCAmelCase	):
    perfect_partitions += 1
  if perfect_partitions > 0:
   if perfect_partitions / total_partitions < max_proportion:
    return int(_lowerCAmelCase	)
  integer += 1


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





