Discussion 4: Recursion

disc04.pdf

Recursion

Many students find this topic challenging. Everything gets easier with practice. Please help each other learn.

Q1: Swipe

Implement swipe, which prints the digits of argument n, one per line, first backward then forward. The left-most digit is printed only once. Do not use while or for or str. (Use recursion, of course!)

First print the first line of the output, then make a recursive call, then print the last line of the output.

Q2: Skip Factorial

Define the base case for the skip_factorial function, which returns the product of every other positive integer, starting with n.

If n is even, then the base case will be 2. If n is odd, then the base case will be 1. Try to write a condition that handles both possibilities.

Q3: Recursive Hailstone

Recall the hailstone function from Homework 1. First, pick a positive integer n as the start. If n is even, divide it by 2. If n is odd, multiply it by 3 and add 1. Repeat this process until n is 1. Complete this recursive version of hailstone that prints out the values of the sequence and returns the number of steps.

An even number is never a base case, so even always makes a recursive call to hailstone and returns one more than the length of the rest of the hailstone sequence.

An odd number might be 1 (the base case) or greater than one (the recursive case). Only the recursive case should call hailstone.

Document the Occasion

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Extra Questions

The questions below are optional but recommended if you would like some extra practice.

Q4: Is Prime

Implement is_prime that takes an integer n greater than 1. It returns True if n is a prime number and False otherwise. Try following the approach below, but implement it recursively without using a while (or for) statement.

def is_prime(n):
    assert n > 1
    i = 2
    while i < n:
        if n % i == 0:
            return False
        i = i + 1
    return True

You will need to define another "helper" function (a function that exists just to help implement this one). Does it matter whether you define it within is_prime or as a separate function in the global frame? Try to define it to take as few arguments as possible.

Define an inner function that checks whether some integer between i and n evenly divides n. Then you can call it starting with i=2:

def is_prime(n):
    def f(i):
        if n % i == 0:
            return ____
        elif ____:
            return ____
        else:
            return f(____)
    return f(2)