
from ..utils import DummyObject, requires_backends







class        lowerCAmelCase__      (   metaclass=__lowercase	):
						UpperCamelCase_							: Any	=		["torch", "scipy"]

						def __init__(			self       ,							*a       ,							**a				)					->      List[Any]:


								'''simple docstring'''

								requires_backends(self       ,							["""torch""", """scipy"""]				)

						@classmethod
						def  A_				(			cls       ,							*a       ,							**a				)					->      Dict:


								'''simple docstring'''

								requires_backends(cls       ,							["""torch""", """scipy"""]				)







						@classmethod
						def  A_				(			cls       ,							*a       ,							**a				)					->      int:


								'''simple docstring'''

								requires_backends(cls       ,							["""torch""", """scipy"""]				)



from __future__ import annotations


def 							__A(lowerCAmelCase						,  lowerCAmelCase						,  lowerCAmelCase						,  lowerCAmelCase						,  lowerCAmelCase						,  )		->				None:

		"""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 							__A(lowerCAmelCase )		->				None:

		"""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)



