def factorial_iterative(n):
    """ Returns the factorial of n, iteratively """
    num = 1
    
    for i in range(2, n+1):
        num *= i  # same as num = num * i
    
    return num

def factorial_iterative2(n):
    """ Returns the factorial of n, iteratively """
    num = 1
    
    for i in range(n):
        num *= (i+1)  # same as num = num * i
    
    return num

def factorial(n):
    # check the base case
    if n == 1:
        return 1
    else:
        # the recursive case 
        return n * factorial(n-1)

def rec_sum(n):
    if n == 0:
        return 0
    else:
        return n + rec_sum(n-1)

def rec_sum_list(some_list):
    if some_list == []:
        return 0
    else:
        return some_list[0] + rec_sum_list(some_list[1:])    
    
def rec_sum_list_print(some_list):
    print("Calling rec_sum_list: " + str(some_list))
    if some_list == []:
        print("base case returning 0")
        return 0
    else:
        answer = some_list[0] + rec_sum_list_print(some_list[1:])
        print(str(some_list) + " returning " + str(answer))
        return answer

def reverse(some_string):
    """ Returns the reverse of the input string """ 
    print("calling " + some_string)
    if some_string == "":
        print("return base ")
        return ""
    else:
        answer = reverse(some_string[1:]) + some_string[0]
        print("returning " + answer)
        return answer 
    
def power(base, exponent):
    """ Returns the base^exponent """
    if exponent == 0:
        return 1
    else:
        return base * power(base, exponent-1)
    
def fast_power(base, exponent):
    """ Returns the base^exponent """
    if exponent == 0:
        return 1
    elif exponent % 2 == 0:
        # b^e = b^(e/2) * b^(e/2)
        temp = fast_power(base, exponent/2)
        return temp * temp
    else:
        return base * fast_power(base, exponent-1)