Homework 1 Solutions

Solution Files

You can find the solutions in hw01.py.

Homework Questions

Q0: Welcome Survey

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Q1: A Plus Abs B

Fill in the blanks in the following function for adding a to the absolute value of b, without calling abs.

from operator import add, sub

def a_plus_abs_b(a, b):
    """Return a+abs(b), but without calling abs.

    >>> a_plus_abs_b(2, 3)
    5
    >>> a_plus_abs_b(2, -3)
    5
    """
    if b < 0:
f = sub
else:
f = add
return f(a, b)

Use Ok to test your code:

python3 ok -q a_plus_abs_b

If b is positive, we add the numbers together. If b is negative, we subtract the numbers. Therefore, we choose the operator add or sub based on the sign of b.

Video walkthrough: https://youtu.be/o9eUNrWTr3I

Q2: Two of Three

Write a function that takes three positive numbers and returns the sum of the squares of the two largest numbers. Use only a single line for the body of the function.

def two_of_three(a, b, c):
    """Return x*x + y*y, where x and y are the two largest members of the
    positive numbers a, b, and c.

    >>> two_of_three(1, 2, 3)
    13
    >>> two_of_three(5, 3, 1)
    34
    >>> two_of_three(10, 2, 8)
    164
    >>> two_of_three(5, 5, 5)
    50
    """
return max(a*a+b*b, a*a+c*c, b*b+c*c) # Alternate solution return a**2 + b**2 + c**2 - min(a, b, c)**2

Hint: Consider using the max or min function:

>>> max(1, 2, 3)
3
>>> min(-1, -2, -3)
-3

Use Ok to test your code:

python3 ok -q two_of_three

We use the fact that if a>b and b>0, then square(a)>square(b). So, we can take the max of the sum of squares of all pairs. The max function can take an arbitrary number of arguments.

Alternatively, we can do the sum of squares of all the numbers. Then we pick the smallest value, and subtract the square of that.

Video walkthrough: https://youtu.be/oPN3OCGGb4M

Q3: Largest Factor

Write a function that takes an integer n that is greater than 1 and returns the largest integer that is smaller than n and evenly divides n.

def largest_factor(n):
    """Return the largest factor of n that is smaller than n.

    >>> largest_factor(15) # factors are 1, 3, 5
    5
    >>> largest_factor(80) # factors are 1, 2, 4, 5, 8, 10, 16, 20, 40
    40
    >>> largest_factor(13) # factor is 1 since 13 is prime
    1
    """
factor = n - 1 while factor > 0: if n % factor == 0: return factor factor -= 1

Hint: To check if b evenly divides a, you can use the expression a % b == 0, which can be read as, "the remainder of dividing a by b is 0."

Use Ok to test your code:

python3 ok -q largest_factor

Iterating from n-1 to 1, we return the first integer that evenly divides n. This is guaranteed to be the largest factor of n.

Video walkthrough: https://youtu.be/pVgxbeL4DHQ

Q4: If Function vs Statement

Let's try to write a function that does the same thing as an if statement.

def if_function(condition, true_result, false_result):
    """Return true_result if condition is a true value, and
    false_result otherwise.

    >>> if_function(True, 2, 3)
    2
    >>> if_function(False, 2, 3)
    3
    >>> if_function(3==2, 3+2, 3-2)
    1
    >>> if_function(3>2, 3+2, 3-2)
    5
    """
    if condition:
        return true_result
    else:
        return false_result

Despite the doctests above, this function actually does not do the same thing as an if statement in all cases. To prove this fact, write functions c, t, and f such that with_if_statement prints the number 2, but with_if_function prints both 1 and 2.

def with_if_statement():
    """
    >>> result = with_if_statement()
    2
    >>> print(result)
    None
    """
    if c():
        return t()
    else:
        return f()

def with_if_function():
    """
    >>> result = with_if_function()
    1
    2
    >>> print(result)
    None
    """
    return if_function(c(), t(), f())

def c():
return False
def t():
print(1)
def f():
print(2)

Hint: If you are having a hard time identifying how an if statement and if_function differ, consider the rules of evaluation for if statements and call expressions.

Use Ok to test your code:

python3 ok -q with_if_statement
python3 ok -q with_if_function

The function with_if_function uses a call expression, which guarantees that all of its operand subexpressions will be evaluated before if_function is applied to the resulting arguments.

Therefore, even if c returns False, the function t will be called. When we call t, we print out 1. Then, when we call f, we will also print 2.

By contrast, with_if_statement will never call t if c returns False. Thus, we will only call f, printing 2.

Q5: Hailstone

Douglas Hofstadter's Pulitzer-prize-winning book, Gödel, Escher, Bach, poses the following mathematical puzzle.

  1. Pick a positive integer n as the start.
  2. If n is even, divide it by 2.
  3. If n is odd, multiply it by 3 and add 1.
  4. Continue this process until n is 1.

The number n will travel up and down but eventually end at 1 (at least for all numbers that have ever been tried -- nobody has ever proved that the sequence will terminate). Analogously, a hailstone travels up and down in the atmosphere before eventually landing on earth.

This sequence of values of n is often called a Hailstone sequence. Write a function that takes a single argument with formal parameter name n, prints out the hailstone sequence starting at n, and returns the number of steps in the sequence:

def hailstone(n):
    """Print the hailstone sequence starting at n and return its
    length.

    >>> a = hailstone(10)
    10
    5
    16
    8
    4
    2
    1
    >>> a
    7
    """
length = 1 while n != 1: print(n) if n % 2 == 0: n = n // 2 # Integer division prevents "1.0" output else: n = 3 * n + 1 length = length + 1 print(n) # n is now 1 return length

Hailstone sequences can get quite long! Try 27. What's the longest you can find?

Use Ok to test your code:

python3 ok -q hailstone

We keep track of the current length of the hailstone sequence and the current value of the hailstone sequence. From there, we loop until we hit the end of the sequence, updating the length in each step.

Note: we need to do floor division // to remove decimals.

Video walkthrough: https://youtu.be/lZZQ0BpsXIc

Extra questions

Extra questions are not worth extra credit and are entirely optional. They are designed to challenge you to think creatively! They do not resemble any exam questions or relate directly to any required content in the course. Feel free to skip them.

Q6: Quine

Write a one-line program that prints itself, using only the following features of the Python language:

  • Number literals
  • Assignment statements
  • String literals that can be expressed using single or double quotes
  • The arithmetic operators +, -, *, and /
  • The built-in print function
  • The built-in eval function, which evaluates a string as a Python expression
  • The built-in repr function, which returns an expression that evaluates to its argument

You can concatenate two strings by adding them together with + and repeat a string by multipying it by an integer. Semicolons can be used to separate multiple statements on the same line. E.g.,

>>> c='c';print('a');print('b' + c * 2)
a
bcc

Hint: Explore the relationship between single quotes, double quotes, and the repr function applied to strings.

A program that prints itself is called a Quine. Place your solution in the multi-line string named quine.

Note: No tests will be run on your solution to this problem.