




from __future__ import annotations





def 		__A		(	__lowerCamelCase							,			__lowerCamelCase    )      ->       list[tuple[int, int]]:
	a				, a										=							position
	a										=							[
	    (y + 1, x + 2),
	    (y - 1, x + 2),
	    (y + 1, x - 2),
	    (y - 1, x - 2),
	    (y + 2, x + 1),
	    (y + 2, x - 1),
	    (y - 2, x + 1),
	    (y - 2, x - 1),
	]
	a										=							[]

	for position in positions:
		a				, a										=							position
		if 0 <= y_test < n and 0 <= x_test < n:
			permissible_positions.append(__lowerCamelCase    )

	return permissible_positions





def 		__A		(	__lowerCamelCase    )      ->       bool:
	return not any(elem == 0 for row in board for elem in row    )





def 		__A		(	__lowerCamelCase							,			__lowerCamelCase							,			__lowerCamelCase    )      ->       bool:

	if is_complete(__lowerCamelCase    ):
		return True

	for position in get_valid_pos(__lowerCamelCase							,			len(__lowerCamelCase    )    ):
		a				, a										=							position

		if board[y][x] == 0:
			a										=							curr + 1
			if open_knight_tour_helper(__lowerCamelCase							,			__lowerCamelCase							,			curr + 1    ):
				return True
			a										=							0

	return False





def 		__A		(	__lowerCamelCase    )      ->       list[list[int]]:
	a										=							[[0 for i in range(__lowerCamelCase    )] for j in range(__lowerCamelCase    )]

	for i in range(__lowerCamelCase    ):
		for j in range(__lowerCamelCase    ):
			a										=							1
			if open_knight_tour_helper(__lowerCamelCase							,			(i, j)							,			1    ):
				return board
			a										=							0

	a										=							f'Open Kight Tour cannot be performed on a board of size {n}'
	raise ValueError(__lowerCamelCase    )


if __name__ == "__main__":
		import doctest

		doctest.testmod()







def 		__A		(	__lowerCamelCase    )      ->       int:
	a										=							0
	while num > 0:
		digit_sum += num % 10
		num //= 10
	return digit_sum





def 		__A		(	__lowerCamelCase = 100    )      ->       int:
	a										=							1
	a										=							2

	for i in range(2							,			max_n + 1    ):
		a										=							pre_numerator
		a										=							2 * i // 3 if i % 3 == 0 else 1
		a										=							cur_numerator
		a										=							e_cont * pre_numerator + temp

	return sum_digits(__lowerCamelCase    )


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

