Homework 3
Due at 11:59:59 pm on Thursday, 9/21/2023.

Instructions

Download hw03.zip. Inside the archive, you will find starter files for the questions in this homework, along with a copy of the OK autograder.

Readings: This homework relies on following references:

• Section 1.6

Lists Review

Question 1: Adding matrices

Write a function that adds two matrices together using list comprehensions. The function should take in two 2D lists of the same dimensions. Try to implement this in one line!

def add_matrices(x, y):
    """
    >>> matrix1 = [[1, 3],
    ...            [2, 0]]
    >>> matrix2 = [[-3, 0],
    ...            [1, 2]]
    >>> add_matrices(matrix1, matrix2)
    [[-2, 3], [3, 2]]
    >>> matrix4 = [[ 1, -2,  3],
    ...            [-4,  5, -6]]
    >>> matrix5 = [[-1,  2, -3],
    ...            [ 4, -5,  6]]
    >>> add_matrices(matrix4, matrix5)
    [[0, 0, 0], [0, 0, 0]]
    """
    "*** YOUR CODE HERE ***"
    

Use OK to test your code:

python3 ok -q add_matrices

Higher Order Functions

Question 2: Mul_by_num

Using higher order functions, complete the mul_by_num function. This function should take an argument and return a one argument function that multiplies any value passed to it by the original number.

def mul_by_num(factor):
    """
    Returns a function that takes one argument and
    returns the product of factor and that argument.
    >>> x = mul_by_num(5)
    >>> y = mul_by_num(2)
    >>> x(3)
    15
    >>> y(-4)
    -8
    """
    "*** YOUR CODE HERE ***"
    

Use OK to test your code:

python3 ok -q mul_by_num

Question 3: This Question is so Derivative

Define a function make_derivative that returns a function: the derivative of a function f. Assuming that f is a single-variable mathematical function, its derivative will also be a single-variable function. When called with a number a, the derivative will estimate the slope of f at point (a, f(a)).

Recall that the formula for finding the derivative of f at point a is:

where h approaches 0. We will approximate the derivative by choosing a very small value for h. The closer h is to 0, the better the estimate of the derivative will be.

def make_derivative(f):
    """Returns a function that approximates the derivative of f.

    Recall that f'(a) = (f(a + h) - f(a)) / h as h approaches 0. We will
    approximate the derivative by choosing a very small value for h.

    >>> def square(x): 
    ...     # equivalent to: square = lambda x: x*x
    ...     return x*x
    >>> derivative = make_derivative(square)
    >>> result = derivative(3)
    >>> round(result, 3) # approximately 2*3
    6.0
    """
    h=0.00001
    "*** YOUR CODE HERE ***"

Use OK to test your code:

python3 ok -q make_derivative

Question 4: Count van Count

Consider the following implementations of count_factors and count_primes:

def count_factors(n):
    """Return the number of positive factors that n has."""
    i, count = 1, 0
    while i <= n:
        if n % i == 0:
            count += 1
        i += 1
    return count

def count_primes(n):
    """Return the number of prime numbers up to and including n."""
    i, count = 1, 0
    while i <= n:
        if is_prime(i):
            count += 1
        i += 1
    return count

def is_prime(n):
    return count_factors(n) == 2 # only factors are 1 and n

The implementations look quite similar! Generalize this logic by writing a function count_cond, which takes in a two-argument predicate function mystery_function(n, i). count_cond returns a count of all the numbers from 1 to n that satisfy mystery_function.

Note: A predicate function is a function that returns a boolean (True or False).

def count_cond(mystery_function, n):
    """
    >>> def divisible(n, i):
    ...     return n % i == 0
    >>> count_cond(divisible, 2) # 1, 2
    2
    >>> count_cond(divisible, 4) # 1, 2, 4
    3
    >>> count_cond(divisible, 12) # 1, 2, 3, 4, 6, 12
    6

    >>> def is_prime(n, i):
    ...     return count_cond(divisible, i) == 2
    >>> count_cond(is_prime, 2) # 2
    1
    >>> count_cond(is_prime, 3) # 2, 3
    2
    >>> count_cond(is_prime, 4) # 2, 3
    2
    >>> count_cond(is_prime, 5) # 2, 3, 5
    3
    >>> count_cond(is_prime, 20) # 2, 3, 5, 7, 11, 13, 17, 19
    8
    """
    "*** YOUR CODE HERE ***"
    

Use OK to test your code:

python3 ok -q count_cond

Question 5: I Heard You Liked Functions...

Define a function cycle that takes in three functions f1, f2, f3, as arguments. cycle will return another function that should take in an integer argument n and return another function. That final function should take in an argument x and cycle through applying f1, f2, and f3 to x, depending on what n was. Here's the what the final function should do to x for a few values of n:

• n = 0, return x
• n = 1, apply f1 to x, or return f1(x)
• n = 2, apply f1 to x and then f2 to the result of that, or return f2(f1(x))
• n = 3, apply f1 to x, f2 to the result of applying f1, and then f3 to the result of applying f2, or f3(f2(f1(x)))
• n = 4, start the cycle again applying f1, then f2, then f3, then f1 again, or f1(f3(f2(f1(x))))
• And so forth.

Hint: most of the work goes inside the most nested function.
Hint 2: given n, how many function calls are made on x?
Hint 3: for help with how to cycle through the functions (i.e., how to go back to applying f1 as your outermost function call when n = 4), consider looking at this python tutor demo which has similar cycling behaviour.

def cycle(f1, f2, f3):
    """ Returns a function that is itself a higher order function
    >>> def add1(x):
    ...     return x + 1
    >>> def times2(x):
    ...     return x * 2
    >>> def add3(x):
    ...     return x + 3
    >>> my_cycle = cycle(add1, times2, add3)
    >>> identity = my_cycle(0)
    >>> identity(5)
    5
    >>> add_one_then_double = my_cycle(2)
    >>> add_one_then_double(1)
    4
    >>> do_all_functions = my_cycle(3)
    >>> do_all_functions(2)
    9
    >>> do_more_than_a_cycle = my_cycle(4)
    >>> do_more_than_a_cycle(2)
    10
    >>> do_two_cycles = my_cycle(6)
    >>> do_two_cycles(1)
    19
    """
    "*** YOUR CODE HERE ***"

Use OK to test your code:

python3 ok -q cycle

Question 6: The Word Guessing Game

Write a higher order function called store_word that takes in a secret word. It will return the length of the secret word and another function, guess_word, that the user can use to try to guess the secret word.

Assume that when the user tries to guess the secret word, they will only guess words that are equal in length to the secret word. The user can pass their guess into the guess_word function, and it will return a list where every element in the list is a boolean, True or False, indicating whether the letter at that index matches the letter in the secret word!

def store_word(secret):
    """
    >>> word_len, guess_word = store_word("cake")
    >>> word_len
    4
    >>> guess_word("corn")
    [True, False, False, False]
    >>> guess_word("come")
    [True, False, False, True]
    >>> guess_word("cake")
    [True, True, True, True]
    >>> word_len, guess_word = store_word("pop")
    >>> word_len
    3
    >>> guess_word("ate")
    [False, False, False]
    >>> guess_word("top")
    [False, True, True]
    >>> guess_word("pop")
    [True, True, True]
    """
    "*** YOUR CODE HERE ***"

Use OK to test your code:

python3 ok -q store_word

Submission

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

• hw03.py