
"""simple docstring"""

import math






def 							A						(    snake_case				:int		)				->     bool:
 __UpperCamelCase      =	math.loga(math.sqrt(4 * positive_integer + 1		) / 2 + 1 / 2		)

 return exponent == int(snake_case		)






def 							A						(    snake_case				:float = 1 / 1_2_3_4_5		)				->     int:
 __UpperCamelCase      =	0
 __UpperCamelCase      =	0

 __UpperCamelCase      =	3
 while True:
  __UpperCamelCase      =	(integer**2 - 1) / 4
  # if candidate is an integer, then there is a partition for k
  if partition_candidate == int(snake_case		):
   __UpperCamelCase      =	int(snake_case		)
   total_partitions += 1
   if check_partition_perfect(snake_case		):
    perfect_partitions += 1
  if perfect_partitions > 0:
   if perfect_partitions / total_partitions < max_proportion:
    return int(snake_case		)
  integer += 1


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





"""simple docstring"""

def 							A						(    snake_case				:list[list[int]]    , snake_case				:int    , snake_case				:int    , snake_case				:list[int]		)				->     bool:

 # 1. Validate that path exists between current and next vertices
 if graph[path[curr_ind - 1]][next_ver] == 0:
  return False

 # 2. Validate that next vertex is not already in path
 return not any(vertex == next_ver for vertex in path		)






def 							A						(    snake_case				:list[list[int]]    , snake_case				:list[int]    , snake_case				:int		)				->     bool:

 # Base Case
 if curr_ind == len(snake_case		):
  # return whether path exists between current and starting vertices
  return graph[path[curr_ind - 1]][path[0]] == 1

 # Recursive Step
 for next_ver in range(0    , len(snake_case		)		):
  if valid_connection(snake_case    , snake_case    , snake_case    , snake_case		):
   # Insert current vertex  into path as next transition
   __UpperCamelCase      =	next_ver
   # Validate created path
   if util_hamilton_cycle(snake_case    , snake_case    , curr_ind + 1		):
    return True
   # Backtrack
   __UpperCamelCase      =	-1
 return False






def 							A						(    snake_case				:list[list[int]]    , snake_case				:int = 0		)				->     list[int]:
 __UpperCamelCase      =	[-1] * (len(snake_case		) + 1)
 # initialize start and end of path with starting index
 __UpperCamelCase      =	__UpperCamelCase      =	start_index
 # evaluate and if we find answer return path either return empty array
 return path if util_hamilton_cycle(snake_case    , snake_case    , 1		) else []



