'''simple docstring'''







from __future__ import annotations


def    lowerCamelCase       (			UpperCAmelCase__				:  Tuple				,						UpperCAmelCase__				:  Optional[int]				,						UpperCAmelCase__				:  Optional[int]				,						UpperCAmelCase__				:  Dict				,						UpperCAmelCase__				:  Union[str, Any]				,						)			->	Tuple:


							'''simple docstring'''






							SCREAMING_SNAKE_CASE__     :Optional[int]						    = len(UpperCAmelCase__		)

							# 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(UpperCAmelCase__		):
														# 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]				,						UpperCAmelCase__				,						UpperCAmelCase__				,						)


def    lowerCamelCase       (			UpperCAmelCase__				:  Optional[Any]		)			->	Optional[int]:


							'''simple docstring'''






							SCREAMING_SNAKE_CASE__     :list[list[str]]						    = []
							depth_first_search([]				,						[]				,						[]				,						UpperCAmelCase__				,						UpperCAmelCase__		)

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

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


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		n_queens_solution(4)














import math
from collections.abc import Iterator
from itertools import takewhile

def   __lowercase       (			snake_case			):







				"""simple docstring"""






				if 1 < number < 4:
								# 2 and 3 are primes
								return True
				elif number < 2 or number % 2 == 0 or number % 3 == 0:
								# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
								return False

				# All primes number are in format of 6k +/- 1
				for i in range(5,	int(math.sqrt(snake_case			) + 1			),	6			):
								if number % i == 0 or number % (i + 2) == 0:
												return False
				return True

def   __lowercase       (			):







				"""simple docstring"""






				__magic_name__		:str   =   2
				while True:
								if is_prime(snake_case			):
												yield num
								num += 1

def   __lowercase       (			snake_case = 2_0_0_0_0_0_0			):







				"""simple docstring"""






				return sum(takewhile(lambda snake_case			: x < n,	prime_generator()			)			)


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






