

from __future__ import annotations


def        __lowercase    (  UpperCAmelCase__		,					UpperCAmelCase__      ):

					"""simple docstring"""






					__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(UpperCAmelCase__      )

					return permissible_positions


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					return not any(elem == 0 for row in board for elem in row      )


def        __lowercase    (  UpperCAmelCase__		,					UpperCAmelCase__		,					UpperCAmelCase__      ):

					"""simple docstring"""






					if is_complete(UpperCAmelCase__      ):
										return True

					for position in get_valid_pos(UpperCAmelCase__		,					len(UpperCAmelCase__      )      ):
										__lowerCAmelCase,	__lowerCAmelCase					=							position

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

					return False


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					__lowerCAmelCase					=							[[0 for i in range(UpperCAmelCase__      )] for j in range(UpperCAmelCase__      )]

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

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


if __name__ == "__main__":
		import doctest

		doctest.testmod()









from __future__ import annotations

from collections.abc import MutableSequence





class     snake_case_			:


							"""simple docstring"""





							def __init__( self					,						_A					,						_A							):
												if len(_A							) != degree + 1:
																	raise ValueError(
																	    'The number of coefficients should be equal to the degree + 1.'							)

												__lowerCAmelCase					=							list(_A							)
												__lowerCAmelCase					=							degree




							def __add__( self					,						_A							):

												if self.degree > polynomial_a.degree:
																	__lowerCAmelCase					=							self.coefficients[:]
																	for i in range(polynomial_a.degree + 1							):
																						coefficients[i] += polynomial_a.coefficients[i]
																	return Polynomial(self.degree					,						_A							)
												else:
																	__lowerCAmelCase					=							polynomial_a.coefficients[:]
																	for i in range(self.degree + 1							):
																						coefficients[i] += self.coefficients[i]
																	return Polynomial(polynomial_a.degree					,						_A							)




							def __sub__( self					,						_A							):
												return self + polynomial_a * Polynomial(0					,						[-1]							)




							def __neg__( self							):
												return Polynomial(self.degree					,						[-c for c in self.coefficients]							)




							def __mul__( self					,						_A							):
												__lowerCAmelCase					=							[0] * (self.degree + polynomial_a.degree + 1)
												for i in range(self.degree + 1							):
																	for j in range(polynomial_a.degree + 1							):
																						coefficients[i + j] += (
																						    self.coefficients[i] * polynomial_a.coefficients[j]
																						)

												return Polynomial(self.degree + polynomial_a.degree					,						_A							)




							def        A__  ( self					,						_A							):
												__lowerCAmelCase					=							0
												for i in range(self.degree + 1							):
																	result += self.coefficients[i] * (substitution**i)
												return result




							def __str__( self							):
												__lowerCAmelCase					=							''
												for i in range(self.degree					,						-1					,						-1							):
																	if self.coefficients[i] == 0:
																						continue
																	elif self.coefficients[i] > 0:
																						if polynomial:
																											polynomial += " + "
																	else:
																						polynomial += " - "

																	if i == 0:
																						polynomial += str(abs(self.coefficients[i]							)							)
																	elif i == 1:
																						polynomial += str(abs(self.coefficients[i]							)							) + "x"
																	else:
																						polynomial += str(abs(self.coefficients[i]							)							) + "x^" + str(_A							)

												return polynomial




							def __repr__( self							):
												return self.__str__()




							def        A__  ( self							):
												__lowerCAmelCase					=							[0] * self.degree
												for i in range(self.degree							):
																	__lowerCAmelCase					=							self.coefficients[i + 1] * (i + 1)
												return Polynomial(self.degree - 1					,						_A							)




							def        A__  ( self					,						_A = 0							):
												__lowerCAmelCase					=							[0] * (self.degree + 2)
												__lowerCAmelCase					=							constant
												for i in range(self.degree + 1							):
																	__lowerCAmelCase					=							self.coefficients[i] / (i + 1)
												return Polynomial(self.degree + 1					,						_A							)




							def __eq__( self					,						_A							):
												if not isinstance(_A					,						_A							):
																	return False

												if self.degree != polynomial_a.degree:
																	return False

												for i in range(self.degree + 1							):
																	if self.coefficients[i] != polynomial_a.coefficients[i]:
																						return False

												return True




							def __ne__( self					,						_A							):
												return not self.__eq__(_A							)






