```
"""
The Josephus problem is a famous theoretical problem related to a certain
counting-out game. This module provides functions to solve the Josephus problem
for num_people and a step_size.
The Josephus problem is defined as follows:
- num_people are standing in a circle.
- Starting with a specified person, you count around the circle,
skipping a fixed number of people (step_size).
- The person at which you stop counting is eliminated from the circle.
- The counting continues until only one person remains.
For more information about the Josephus problem, refer to:
https://en.wikipedia.org/wiki/Josephus_problem
"""
def josephus_recursive(num_people: int, step_size: int) -> int:
"""
Solve the Josephus problem for num_people and a step_size recursively.
Args:
num_people: A positive integer representing the number of people.
step_size: A positive integer representing the step size for elimination.
Returns:
The position of the last person remaining.
Raises:
ValueError: If num_people or step_size is not a positive integer.
Examples:
>>> josephus_recursive(7, 3)
3
>>> josephus_recursive(10, 2)
4
>>> josephus_recursive(0, 2)
Traceback (most recent call last):
...
ValueError: num_people or step_size is not a positive integer.
>>> josephus_recursive(1.9, 2)
Traceback (most recent call last):
...
ValueError: num_people or step_size is not a positive integer.
>>> josephus_recursive(-2, 2)
Traceback (most recent call last):
...
ValueError: num_people or step_size is not a positive integer.
>>> josephus_recursive(7, 0)
Traceback (most recent call last):
...
ValueError: num_people or step_size is not a positive integer.
>>> josephus_recursive(7, -2)
Traceback (most recent call last):
...
ValueError: num_people or step_size is not a positive integer.
>>> josephus_recursive(1_000, 0.01)
Traceback (most recent call last):
...
ValueError: num_people or step_size is not a positive integer.
>>> josephus_recursive("cat", "dog")
Traceback (most recent call last):
...
ValueError: num_people or step_size is not a positive integer.
"""
if (
not isinstance(num_people, int)
or not isinstance(step_size, int)
or num_people <= 0
or step_size <= 0
):
raise ValueError("num_people or step_size is not a positive integer.")
if num_people == 1:
return 0
return (josephus_recursive(num_people - 1, step_size) + step_size) % num_people
def find_winner(num_people: int, step_size: int) -> int:
"""
Find the winner of the Josephus problem for num_people and a step_size.
Args:
num_people (int): Number of people.
step_size (int): Step size for elimination.
Returns:
int: The position of the last person remaining (1-based index).
Examples:
>>> find_winner(7, 3)
4
>>> find_winner(10, 2)
5
"""
return josephus_recursive(num_people, step_size) + 1
def josephus_iterative(num_people: int, step_size: int) -> int:
"""
Solve the Josephus problem for num_people and a step_size iteratively.
Args:
num_people (int): The number of people in the circle.
step_size (int): The number of steps to take before eliminating someone.
Returns:
int: The position of the last person standing.
Examples:
>>> josephus_iterative(5, 2)
3
>>> josephus_iterative(7, 3)
4
"""
circle = list(range(1, num_people + 1))
current = 0
while len(circle) > 1:
current = (current + step_size - 1) % len(circle)
circle.pop(current)
return circle[0]
if __name__ == "__main__":
import doctest
doctest.testmod()
```