The Josephus problem is a famous theoretical problem related to a certain countingout game. This module provides functions to solve the Josephus problem for numpeople and a stepsize. The Josephus problem is defined as follows:  numpeople are standing in a circle.  Starting with a specified person, you count around the circle, skipping a fixed number of people stepsize.  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.orgwikiJosephusproblem Solve the Josephus problem for numpeople and a stepsize recursively. Args: numpeople: A positive integer representing the number of people. stepsize: A positive integer representing the step size for elimination. Returns: The position of the last person remaining. Raises: ValueError: If numpeople or stepsize is not a positive integer. Examples:  josephusrecursive7, 3 3  josephusrecursive10, 2 4  josephusrecursive0, 2 Traceback most recent call last: ... ValueError: numpeople or stepsize is not a positive integer.  josephusrecursive1.9, 2 Traceback most recent call last: ... ValueError: numpeople or stepsize is not a positive integer.  josephusrecursive2, 2 Traceback most recent call last: ... ValueError: numpeople or stepsize is not a positive integer.  josephusrecursive7, 0 Traceback most recent call last: ... ValueError: numpeople or stepsize is not a positive integer.  josephusrecursive7, 2 Traceback most recent call last: ... ValueError: numpeople or stepsize is not a positive integer.  josephusrecursive1000, 0.01 Traceback most recent call last: ... ValueError: numpeople or stepsize is not a positive integer.  josephusrecursivecat, dog Traceback most recent call last: ... ValueError: numpeople or stepsize is not a positive integer. Find the winner of the Josephus problem for numpeople and a stepsize. Args: numpeople int: Number of people. stepsize int: Step size for elimination. Returns: int: The position of the last person remaining 1based index. Examples:  findwinner7, 3 4  findwinner10, 2 5 Solve the Josephus problem for numpeople and a stepsize iteratively. Args: numpeople int: The number of people in the circle. stepsize int: The number of steps to take before eliminating someone. Returns: int: The position of the last person standing. Examples:  josephusiterative5, 2 3  josephusiterative7, 3 4
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()
