


'''simple docstring'''






from __future__ import annotations


def  _snake_case       (					A			,   A	)						->   list[tuple[int, int]]:
							lowerCAmelCase__       , lowerCAmelCase__	       =     position
							lowerCAmelCase__	       =     [
							    (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),
							]
							lowerCAmelCase__	       =     []

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

							return permissible_positions


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


def  _snake_case       (					A			,   A			,   A	)						->   bool:

							if is_complete(A	):
														return True

							for position in get_valid_pos(A			,   len(A	)	):
														lowerCAmelCase__       , lowerCAmelCase__	       =     position

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

							return False


def  _snake_case       (					A	)						->   list[list[int]]:
							lowerCAmelCase__	       =     [[0 for i in range(A	)] for j in range(A	)]

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

							lowerCAmelCase__	       =     F"""Open Kight Tour cannot be performed on a board of size {n}"""
							raise ValueError(A	)


if __name__ == "__main__":
		import doctest

		doctest.testmod()



'''simple docstring'''






from __future__ import annotations

import math
from collections.abc import Callable


def  _snake_case       (					A			,   A			,   A			,   A = 100			,   )						->   float:
							lowerCAmelCase__	       =     x_start
							lowerCAmelCase__	       =     fnc(A	)
							lowerCAmelCase__	       =     0.0

							for _ in range(A	):
														# Approximates curve as a sequence of linear lines and sums their length
														lowerCAmelCase__	       =     (x_end - x_start) / steps + xa
														lowerCAmelCase__	       =     fnc(A	)
														length += math.hypot(xa - xa			,   fxa - fxa	)

														# Increment step
														lowerCAmelCase__	       =     xa
														lowerCAmelCase__	       =     fxa

							return length


if __name__ == "__main__":


		def  _snake_case       (					A	)						->   List[Any]:
									return math.sin(10 * x	)

		print('''f(x) = sin(10 * x)''')
		print('''The length of the curve from x = -10 to x = 10 is:''')
		__UpperCAmelCase       =							10
		while i <= 100_000:
				print(f"""With {i} steps: {line_length(f, -10, 10, i)}""")
				i *= 10