


from __future__ import annotations

import numpy as np
def 				_UpperCAmelCase							(UpperCamelCase__	:    np.ndarray		):
       _A       ,     _A :			List[Any]							       =					np.shape(snake_case__		)
       if rows != columns:
              _A :			Tuple							       =					(
                  "\'table\' has to be of square shaped array but got a "
                  f"{rows}x{columns} array:\n{table}"
              )
              raise ValueError(snake_case__		)

       _A :			str							       =					np.zeros((rows, columns)		)
       _A :			Any							       =					np.zeros((rows, columns)		)
       for i in range(snake_case__		):
              for j in range(snake_case__		):
                     _A :			List[Any]							       =					sum(lower[i][k] * upper[k][j] for k in range(snake_case__		)		)
                     if upper[j][j] == 0:
                            raise ArithmeticError("No LU decomposition exists"		)
                     _A :			int							       =					(table[i][j] - total) / upper[j][j]
              _A :			Optional[Any]							       =					1
              for j in range(snake_case__							,				snake_case__		):
                     _A :			List[str]							       =					sum(lower[i][k] * upper[k][j] for k in range(snake_case__		)		)
                     _A :			int							       =					table[i][j] - total
       return lower, upper


if __name__ == "__main__":
    import doctest

    doctest.testmod()


"""simple docstring"""


from __future__ import annotations






def    _snake_case					(		snake_case__ :  tuple[int, int]						,   snake_case__ :  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(snake_case__  )

	return permissible_positions






def    _snake_case					(		snake_case__ :  list[list[int]]  ):
	return not any(elem == 0 for row in board for elem in row  )






def    _snake_case					(		snake_case__ :  list[list[int]]						,   snake_case__ :  tuple[int, int]						,   snake_case__ :  int  ):

	if is_complete(snake_case__  ):
		return True

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

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

	return False






def    _snake_case					(		snake_case__ :  int  ):
	A     = [[0 for i in range(snake_case__  )] for j in range(snake_case__  )]

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

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


if __name__ == "__main__":
		import doctest

		doctest.testmod()