

"""simple docstring"""






from itertools import permutations
def       UpperCamelCase					(			_lowerCAmelCase     :						Optional[int] = 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
	_UpperCAmelCase    :							str									=					list(range(gon_side + 1,  0,  -1 ) )
	_UpperCAmelCase    :							Dict									=					list(range(gon_side + 2,  gon_side * 2 + 1 ) )

	for perm in permutations(small_numbers + big_numbers ):
		_UpperCAmelCase    :							Any									=					generate_gon_ring(__a,  list(__a ) )
		if is_magic_gon(__a ):
			return int("""""".join(str(__a ) for n in numbers ) )

	_UpperCAmelCase    :							Optional[Any]									=					f'''Magic {gon_side}-gon ring is impossible'''
	raise ValueError(__a )
def       UpperCamelCase					(			_lowerCAmelCase     :						List[Any],  _lowerCAmelCase     :						Dict )				->			list[int]:
	_UpperCAmelCase    :							Union[str, Any]									=					[0] * (gon_side * 3)
	_UpperCAmelCase    :							str									=					perm[0:3]
	perm.append(perm[1] )

	_UpperCAmelCase    :							List[str]									=					1 if gon_side < 5 else 2

	for i in range(1,  len(__a ) // 3 + magic_number ):
		_UpperCAmelCase    :							Optional[int]									=					perm[2 * i + 1]
		_UpperCAmelCase    :							List[str]									=					result[3 * i - 1]
		_UpperCAmelCase    :							int									=					perm[2 * i + 2]

	return result
def       UpperCamelCase					(			_lowerCAmelCase     :						Optional[int] )				->			bool:





	if len(__a ) % 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

	_UpperCAmelCase    :							Optional[int]									=					sum(numbers[:3] )

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


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






"""simple docstring"""






def       UpperCamelCase					(			_lowerCAmelCase     :						int,  _lowerCAmelCase     :						int = 1000 )				->			str:
	if (not isinstance(_lowerCAmelCase,  _lowerCAmelCase )) or (digit_position <= 0):
		raise ValueError("""Digit position must be a positive integer""" )
	elif (not isinstance(_lowerCAmelCase,  _lowerCAmelCase )) or (precision < 0):
		raise ValueError("""Precision must be a nonnegative integer""" )

	# compute an approximation of (16 ** (n - 1)) * pi whose fractional part is mostly
	# accurate
	_UpperCAmelCase    :							List[Any]									=					(
	    4 * _subsum(_lowerCAmelCase,  1,  _lowerCAmelCase )
	    - 2 * _subsum(_lowerCAmelCase,  4,  _lowerCAmelCase )
	    - _subsum(_lowerCAmelCase,  5,  _lowerCAmelCase )
	    - _subsum(_lowerCAmelCase,  6,  _lowerCAmelCase )
	)

	# return the first hex digit of the fractional part of the result
	return hex(int((sum_result % 1) * 16 ) )[2:]
def       UpperCamelCase					(			_lowerCAmelCase     :						int,  _lowerCAmelCase     :						int,  _lowerCAmelCase     :						int )				->			float:
	_UpperCAmelCase    :							Optional[int]									=					0.0
	for sum_index in range(digit_pos_to_extract + precision ):
		_UpperCAmelCase    :							Optional[Any]									=					8 * sum_index + denominator_addend
		if sum_index < digit_pos_to_extract:
			# if the exponential term is an integer and we mod it by the denominator
			# before dividing, only the integer part of the sum will change;
			# the fractional part will not
			_UpperCAmelCase    :							Optional[Any]									=					pow(
			    16,  digit_pos_to_extract - 1 - sum_index,  _lowerCAmelCase )
		else:
			_UpperCAmelCase    :							int									=					pow(16,  digit_pos_to_extract - 1 - sum_index )
		total += exponential_term / denominator
	return total


if __name__ == "__main__":
						import doctest

						doctest.testmod()





