
from itertools import permutations
def 					lowerCAmelCase__     (    SCREAMING_SNAKE_CASE_ = 5	)							->						int:
			if gon_side < 3 or gon_side > 5:
						raise ValueError('gon_side must be in the range [3, 5]'	)

			# Since it's 16, we know 10 is on the outer ring
			# Put the big numbers at the end so that they are never the first number
			lowerCAmelCase__ : Optional[int]   =						list(range(gon_side + 1			,				0			,				-1	)	)
			lowerCAmelCase__ : Optional[int]   =						list(range(gon_side + 2			,				gon_side * 2 + 1	)	)

			for perm in permutations(small_numbers + big_numbers	):
						lowerCAmelCase__ : List[Any]   =						generate_gon_ring(SCREAMING_SNAKE_CASE_			,				list(SCREAMING_SNAKE_CASE_	)	)
						if is_magic_gon(SCREAMING_SNAKE_CASE_	):
									return int(''.join(str(SCREAMING_SNAKE_CASE_	) for n in numbers	)	)

			lowerCAmelCase__ : str   =						F'''Magic {gon_side}-gon ring is impossible'''
			raise ValueError(SCREAMING_SNAKE_CASE_	)
def 					lowerCAmelCase__     (    SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_	)							->						list[int]:
			lowerCAmelCase__ : List[Any]   =						[0] * (gon_side * 3)
			lowerCAmelCase__ : Dict   =						perm[0:3]
			perm.append(perm[1]	)

			lowerCAmelCase__ : Dict   =						1 if gon_side < 5 else 2

			for i in range(1			,				len(SCREAMING_SNAKE_CASE_	) // 3 + magic_number	):
						lowerCAmelCase__ : Tuple   =						perm[2 * i + 1]
						lowerCAmelCase__ : Union[str, Any]   =						result[3 * i - 1]
						lowerCAmelCase__ : Any   =						perm[2 * i + 2]

			return result
def 					lowerCAmelCase__     (    SCREAMING_SNAKE_CASE_	)							->						bool:
			if len(SCREAMING_SNAKE_CASE_	) % 3 != 0:
						raise ValueError('a gon ring should have a length that is a multiple of 3'	)

			if min(numbers[::3]	) != numbers[0]:
						return False

			lowerCAmelCase__ : str   =						sum(numbers[:3]	)

			return all(sum(numbers[i : i + 3]	) == total for i in range(3			,				len(SCREAMING_SNAKE_CASE_	)			,				3	)	)


if __name__ == "__main__":
					print(solution())

from typing import Any

import numpy as np
def 					lowerCAmelCase__     (    SCREAMING_SNAKE_CASE_	)							->						bool:
			assert np.shape(SCREAMING_SNAKE_CASE_	)[0] == np.shape(SCREAMING_SNAKE_CASE_	)[1]

			# If matrix not symmetric, exit right away.
			if np.allclose(SCREAMING_SNAKE_CASE_			,				matrix.T	) is False:
						return False

			# Get eigenvalues and eignevectors for a symmetric matrix.
			lowerCAmelCase__ ,							lowerCAmelCase__ : Optional[int]   =						np.linalg.eigh(SCREAMING_SNAKE_CASE_	)

			# Check sign of all eigenvalues.
			# np.all returns a value of type np.bool_
			return bool(np.all(eigen_values > 0	)	)
def 					lowerCAmelCase__     (    SCREAMING_SNAKE_CASE_	)							->						Any:
			lowerCAmelCase__ : List[Any]   =						np.random.randn(SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_	)
			lowerCAmelCase__ : Any   =						np.dot(SCREAMING_SNAKE_CASE_			,				random_matrix.T	)
			assert _is_matrix_spd(SCREAMING_SNAKE_CASE_	)
			return spd_matrix
def 					lowerCAmelCase__     (    SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_ = 1_000			,				SCREAMING_SNAKE_CASE_ = 1e-8			,				)							->						Any:
			assert np.shape(SCREAMING_SNAKE_CASE_	)[0] == np.shape(SCREAMING_SNAKE_CASE_	)[1]
			assert np.shape(SCREAMING_SNAKE_CASE_	)[0] == np.shape(SCREAMING_SNAKE_CASE_	)[0]
			assert _is_matrix_spd(SCREAMING_SNAKE_CASE_	)

			# Initialize solution guess, residual, search direction.
			lowerCAmelCase__ : Optional[Any]   =						np.zeros((np.shape(SCREAMING_SNAKE_CASE_	)[0], 1)	)
			lowerCAmelCase__ : str   =						np.copy(SCREAMING_SNAKE_CASE_	)
			lowerCAmelCase__ : Dict   =						np.copy(SCREAMING_SNAKE_CASE_	)

			# Set initial errors in solution guess and residual.
			lowerCAmelCase__ : Union[str, Any]   =						1e9
			lowerCAmelCase__ : int   =						1e9
			lowerCAmelCase__ : Union[str, Any]   =						1e9

			# Set iteration counter to threshold number of iterations.
			lowerCAmelCase__ : List[str]   =						0

			while error > tol:
						# Save this value so we only calculate the matrix-vector product once.
						lowerCAmelCase__ : int   =						np.dot(SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_	)

						# The main algorithm.

						# Update search direction magnitude.
						lowerCAmelCase__ : List[str]   =						np.dot(ra.T			,				SCREAMING_SNAKE_CASE_	) / np.dot(pa.T			,				SCREAMING_SNAKE_CASE_	)
						# Update solution guess.
						lowerCAmelCase__ : Dict   =						xa + alpha * pa
						# Calculate new residual.
						lowerCAmelCase__ : Tuple   =						ra - alpha * w
						# Calculate new Krylov subspace scale.
						lowerCAmelCase__ : str   =						np.dot(r.T			,				SCREAMING_SNAKE_CASE_	) / np.dot(ra.T			,				SCREAMING_SNAKE_CASE_	)
						# Calculate new A conjuage search direction.
						lowerCAmelCase__ : Dict   =						r + beta * pa

						# Calculate errors.
						lowerCAmelCase__ : List[str]   =						np.linalg.norm(r - ra	)
						lowerCAmelCase__ : Tuple   =						np.linalg.norm(x - xa	)
						lowerCAmelCase__ : str   =						np.maximum(SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_	)

						# Update variables.
						lowerCAmelCase__ : List[Any]   =						np.copy(SCREAMING_SNAKE_CASE_	)
						lowerCAmelCase__ : Dict   =						np.copy(SCREAMING_SNAKE_CASE_	)
						lowerCAmelCase__ : str   =						np.copy(SCREAMING_SNAKE_CASE_	)

						# Update number of iterations.
						iterations += 1
						if iterations > max_iterations:
									break

			return x
def 					lowerCAmelCase__     (    )							->						None:
			lowerCAmelCase__ : Optional[Any]   =						3
			lowerCAmelCase__ : Optional[int]   =						_create_spd_matrix(SCREAMING_SNAKE_CASE_	)
			lowerCAmelCase__ : Tuple   =						np.random.randn(SCREAMING_SNAKE_CASE_			,				1	)
			lowerCAmelCase__ : int   =						np.dot(SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_	)

			# Numpy solution.
			lowerCAmelCase__ : Optional[Any]   =						np.linalg.solve(SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_	)

			# Our implementation.
			lowerCAmelCase__ : Optional[int]   =						conjugate_gradient(SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_	)

			# Ensure both solutions are close to x_true (and therefore one another).
			assert np.linalg.norm(x_numpy - x_true	) <= 1e-6
			assert np.linalg.norm(x_conjugate_gradient - x_true	) <= 1e-6


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					test_conjugate_gradient()