def Josephus(n):
    '''
    Function that returns the position of the last remaining person in the
    circle. n is the number of people in the circle (people in the circle
    are numbered from 1 to n), every second person gets killed.
    '''
    circle = [i + 1 for i in range(n)] #Creating a circle of n people numbered from 1 to n.
    i = 0
    while len(circle) != 1:
        i = (i + 1) % len(circle) # .pop automatically skips one for us so we need to skip one less. 
                                  # we use % so that the position in the cicle doesn't goes above the number of people in the circle.
        circle.pop(i)
    return circle[0]

for n in range(1,21):
    print(n, Josephus(n))

def orderJosephus(n):
    '''
    Function that returns the order of killing and position of the last remaining
    person in the circle. n is the number of people in the circle (people in the
    circle are numbered from 1 to n), every second person gets killed.
    '''
    circle = [i + 1 for i in range(n)]
    order_of_killing = [] # empty list to add the removed positions from the list circle.
    i = 0
    while len(circle) != 1:
        i = (i + 1) % len(circle)
        order_of_killing.append(circle.pop(i))
    return order_of_killing, circle[0]

orderJosephus(8)

import math
def j(n):
    '''
    A function that obtains the position of the last survivor using a formula.
    '''
    return 1 + 2 * n - 2 ** (1 + math.floor(math.log(n, 2)))

checks = [j(n) == Josephus(n) for n in range(1,2000)]
all(checks)

def Josephusk(n, k):
    '''
    Function that returns the killing order and position of the last remaining person in the circle. 
    There are n people in the circle (people in the circle are numbered from 1 to n), 
    every kth person is killed until there is only oneperson left.
    '''
    circle = [i + 1 for i in range(n)]
    order_of_killing = []
    i = 0
    while len(circle) != 1:
        i = (i + k - 1) % len(circle) # .pop automatically skips one for us so we need to skip one less. 
        #we use % so that the position never goes above the number of people in the circle
        order_of_killing.append(circle.pop(i)) #create a list of objects removed from circle
    return circle[0]

def phus(n, k):
    '''
    A function that returns the position of survivor using recursion.
    '''
    if n == 1:
        return 1
    return ((phus(n - 1, k) + k - 1) % n) + 1

check = [phus(n, k) == Josephusk(n, k) for n in range(1,200) for k in range(1,200)]
all(check)