
from __future__ import annotations







def     lowerCAmelCase_					( __lowerCamelCase							,     __lowerCamelCase     ):
							__snake_case			,      __snake_case		:      str										=       position
							__snake_case		:      Union[str, Any]										=       [
							    (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),
							]
							__snake_case		:      List[Any]										=       []

							for position in positions:
														__snake_case			,      __snake_case		:      str										=       position
														if 0 <= y_test < n and 0 <= x_test < n:
																					permissible_positions.append(a__     )

							return permissible_positions







def     lowerCAmelCase_					( __lowerCamelCase     ):
							return not any(elem == 0 for row in board for elem in row     )







def     lowerCAmelCase_					( __lowerCamelCase							,     __lowerCamelCase							,     __lowerCamelCase     ):

							if is_complete(a__     ):
														return True

							for position in get_valid_pos(a__							,     len(a__     )     ):
														__snake_case			,      __snake_case		:      Optional[int]										=       position

														if board[y][x] == 0:
																					__snake_case		:      Optional[int]										=       curr + 1
																					if open_knight_tour_helper(a__							,     a__							,     curr + 1     ):
																												return True
																					__snake_case		:      Optional[Any]										=       0

							return False







def     lowerCAmelCase_					( __lowerCamelCase     ):
							__snake_case		:      Any										=       [[0 for i in range(a__     )] for j in range(a__     )]

							for i in range(a__     ):
														for j in range(a__     ):
																					__snake_case		:      Optional[Any]										=       1
																					if open_knight_tour_helper(a__							,     (i, j)							,     1     ):
																												return board
																					__snake_case		:      Tuple										=       0

							__snake_case		:      Union[str, Any]										=       F'Open Kight Tour cannot be performed on a board of size {n}'
							raise ValueError(a__     )


if __name__ == "__main__":
			import doctest

			doctest.testmod()





def     lowerCAmelCase_					( __lowerCamelCase							,     __lowerCamelCase     ):
							return x if y == 0 else greatest_common_divisor(__lowerCamelCase							,     x % y     )







def     lowerCAmelCase_					( __lowerCamelCase							,     __lowerCamelCase     ):
							return (x * y) // greatest_common_divisor(__lowerCamelCase							,     __lowerCamelCase     )







def     lowerCAmelCase_					( __lowerCamelCase = 2_0     ):
							__snake_case		:      Optional[Any]										=       1
							for i in range(1							,     n + 1     ):
														__snake_case		:      Any										=       lcm(__lowerCamelCase							,     __lowerCamelCase     )
							return g


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

