Starter Files

Download lab06.zip. Inside the archive, you will find starter files for the questions in this lab, along with a copy of the OK autograder.

Recursion

Question 1: Sum

Using recursion, write a function sum that takes a single argument n and computes the sum of all integers between 0 and n inclusive. Do not write this function using a while or for loop. Assume n is positive.

def sum(n):
    """Using recursion, computes the sum of all integers between 1 and n, inclusive.
    Assume n is positive.

    >>> sum(1)
    1
    >>> sum(5)  # 1 + 2 + 3 + 4 + 5
    15
    >>> sum(11) # 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11
    66
    """
    "*** YOUR CODE HERE ***"
    if n == 1:
        return 1
    return n + sum(n - 1)

Toggle Solution

Use OK to test your code:

python3 ok -q sum

Question 2: Has Seven

Write a function has_seven that takes a positive integer n and returns whether n contains the digit 7. Do not use any assignment statements - use recursion instead:

def has_seven(k):
    """Returns True if at least one of the digits of k is a 7, False otherwise.

    >>> has_seven(3)
    False
    >>> has_seven(7)
    True
    >>> has_seven(2734)
    True
    >>> has_seven(2634)
    False
    >>> has_seven(734)
    True
    >>> has_seven(7777)
    True
    """
    "*** YOUR CODE HERE ***"
    if k == 0:
        return False
    if k % 10 == 7:
        return True
    else:
        return has_seven(k // 10)

Toggle Solution

Use OK to test your code:

python3 ok -q has_seven

Question 3: Filter

Write the recursive version of the function filter which returns a list and takes in

• f - a one-argument function that returns True if the passed in argument should be included in the resulting list or False otherwise
• seq - a list of values

Note that this is different from the built in filter function we learned previously, which returns a filter object, not a list.

def filter(f, seq):
    """Filter a sequence to only contain values allowed by filter.

    >>> def is_even(x):
    ...     return x % 2 == 0
    >>> def divisible_by5(x):
    ...     return x % 5 == 0
    >>> filter(is_even, [1,2,3,4])
    [2, 4]
    >>> filter(divisible_by5, [1, 4, 9, 16, 25, 100])
    [25, 100]
    >>> filter(is_even, [1])
    []
    >>> filter(is_even, [2])
    [2]
    >>> filter(is_even, [])
    []
    """
    "*** YOUR CODE HERE ***"
    
    if seq == []:
        return seq
    if f(seq[0]):
        return [seq[0]] + filter(f, seq[1:])
    return filter(f, seq[1:])

Toggle Solution

Use OK to test your code:

python3 ok -q filter

Question 4: Decimal

Write the recursive version of the function decimal which takes in an integer n and returns a list of its digits, the decimal representation of n. See the doctests to handle the case where n < 0.

def decimal(n):
    """Return a list representing the decimal representation of a number.

    >>> decimal(2)
    [2]
    >>> decimal(-8)
    ['-', 8]
    >>> decimal(0)
    [0]
    >>> decimal(55055)
    [5, 5, 0, 5, 5]
    >>> decimal(-136)
    ['-', 1, 3, 6]
    """
    "*** YOUR CODE HERE ***"
    if n < 0:
        return ['-'] + decimal(-1 * n)
    elif n < 10:
        return [n]
    else:
        return decimal(n // 10) + [n % 10]

Toggle Solution

Use OK to test your code:

python3 ok -q decimal

Question 5: Insect Combinatorics

Consider an insect in an M by N grid. The insect starts at the bottom left corner, (0, 0), and wants to end up at the top right corner, (M-1, N-1). The insect is only capable of moving right or up. Write a function paths that takes a grid length and width and returns the number of different paths the insect can take from the start to the goal. (There is a closed-form solution to this problem, but try to answer it procedurally using recursion.)

For example, the 2 by 2 grid has a total of two ways for the insect to move from the start to the goal. For the 3 by 3 grid, the insect has 6 different paths (only 3 are shown above). Note that this problem uses tree recursion.

def paths(m, n):
    """Return the number of paths from one corner of an
    M by N grid to the opposite corner.

    >>> paths(2, 2)
    2
    >>> paths(5, 7)
    210
    >>> paths(117, 1)
    1
    >>> paths(1, 157)
    1
    """
    "*** YOUR CODE HERE ***"
    if m == 0 or n == 0:
        return 0
    if m == 1 and n == 1:
        return 1
    return paths(m - 1, n) + paths(m, n - 1)

Toggle Solution

Use OK to test your code:

python3 ok -q paths

Submission

When you are done, submit your file to Gradescope. You only need to upload the following files:

• lab06.py

You may submit more than once before the deadline; only the final submission will be graded. It is your responsibility to check that the autograder on Gradescope runs as expected after you upload your submission.