
"""simple docstring"""

from __future__ import annotations
def 		UpperCAmelCase (		a_,						a_,						a_,						a_,						a_,						):





		'''simple docstring'''


		lowerCamelCase					:      Optional[int]          =   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_      ):





		'''simple docstring'''


		lowerCamelCase					:      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_,						a_      ):





		'''simple docstring'''


		while b:
				lowerCamelCase     ,   lowerCamelCase					:      Tuple          =   b, a % b
		return a
def 		UpperCAmelCase (		a_,						a_      ):





		'''simple docstring'''


		return a if b == 0 else euclidean_gcd_recursive(a_,						a % b      )
def 		UpperCAmelCase (		):





		'''simple docstring'''


		print(F"""euclidean_gcd(3, 5) = {euclidean_gcd(3,						5      )}"""      )
		print(F"""euclidean_gcd(5, 3) = {euclidean_gcd(5,						3      )}"""      )
		print(F"""euclidean_gcd(1, 3) = {euclidean_gcd(1,						3      )}"""      )
		print(F"""euclidean_gcd(3, 6) = {euclidean_gcd(3,						6      )}"""      )
		print(F"""euclidean_gcd(6, 3) = {euclidean_gcd(6,						3      )}"""      )

		print(F"""euclidean_gcd_recursive(3, 5) = {euclidean_gcd_recursive(3,						5      )}"""      )
		print(F"""euclidean_gcd_recursive(5, 3) = {euclidean_gcd_recursive(5,						3      )}"""      )
		print(F"""euclidean_gcd_recursive(1, 3) = {euclidean_gcd_recursive(1,						3      )}"""      )
		print(F"""euclidean_gcd_recursive(3, 6) = {euclidean_gcd_recursive(3,						6      )}"""      )
		print(F"""euclidean_gcd_recursive(6, 3) = {euclidean_gcd_recursive(6,						3      )}"""      )


if __name__ == "__main__":
					main()






