




'''simple docstring'''






from __future__ import annotations

def 			__lowerCAmelCase					(    snake_case__					,      snake_case__					,      snake_case__					,      snake_case__					,      snake_case__					,      ):
			__UpperCamelCase      :			List[Any]									=   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 			__lowerCAmelCase					(    snake_case__						):
			__UpperCamelCase      :			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)









from __future__ import annotations




def 					lowerCamelCase__       (						A__       :					list[int]  ,			A__       :					list[int]  ,			A__       :					list[int]  ,			A__       :					list[list[str]]  ,			A__       :					int  ,			):





       '''simple docstring'''



       __lowerCamelCase   =						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 					lowerCamelCase__       (						A__       :					int	):





       '''simple docstring'''



       __lowerCamelCase   =						[]
       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)

