Discussion 2: Control🖨️

While and If

Learning to use if and while is essential. During this discussion, focus on what we've studied in the first three lectures: if, while, assignment (=), comparison (<, >, ==, ...), and arithmetic. Please don't use any features of Python that we haven't discussed in class yet, such as for, range, and lists. We'll have plenty of time for those later, but now is the time to practice if (textbook 1.5.4) and while (textbook 1.5.5).

Q1: Fizzbuzz

Implement the classic Fizz Buzz sequence. The fizzbuzz function takes a positive integer n and prints out a single line for each integer from 1 to n. For each i:

• If i is divisible by both 3 and 5, print fizzbuzz.
• If i is divisible by 3 (but not 5), print fizz.
• If i is divisible by 5 (but not 3), print buzz.
• Otherwise, print the number i.

Try to make your implementation of fizzbuzz concise.

def fizzbuzz(n):
    """
    >>> result = fizzbuzz(16)
    1
    2
    fizz
    4
    buzz
    fizz
    7
    8
    fizz
    buzz
    11
    fizz
    13
    14
    fizzbuzz
    16
    >>> print(result)
    None
    """
    "*** YOUR CODE HERE ***"

Problem Solving

A useful approach to implementing a function is to:

1. Pick an example input and corresponding output.
2. Describe a process (in English) that computes the output from the input using simple steps.
3. Figure out what additional names you'll need to carry out this process.
4. Implement the process in code using those additional names.
5. Determine whether the implementation really works on your original example.
6. Determine whether the implementation really works on other examples. (If not, you might need to revise step 2.)

Importantly, this approach doesn't go straight from reading a question to writing code.

Important: Don't check your work using a computer right away. Instead, talk to your group and think to try to figure out if an answer is correct. On exams, you won't be able to guess and check because you won't have a Python interpreter. Now is a great time to practice checking your work by thinking through examples. You could even draw an environment diagram!

If you're not sure about how something works or get stuck, ask for help from the course staff.

Q2: Is Prime?

Write a function that returns True if a positive integer n is a prime number and False otherwise.

A prime number n is a number that is not divisible by any numbers other than 1 and n itself. For example, 13 is prime, since it is only divisible by 1 and 13, but 14 is not, since it is divisible by 1, 2, 7, and 14.

Use the % operator: x % y returns the remainder of x when divided by y.

The approach to implementing functions described above would go:

1. Pick n is 9 as the input and False as the output.
2. Invent a process to determine whether 9 is prime, such as: Check that 9 (n) is not a multiple of any integers between 1 and 9 (n).
3. Introduce i to represent each number between 1 and 9 (n).
4. Implement is_prime
5. Check that is_prime(9) will return False by thinking through the execution of the code.
6. Check that is_prime(3) will return True and is_prime(1) will return False.

def is_prime(n):
    """
    >>> is_prime(10)
    False
    >>> is_prime(7)
    True
    >>> is_prime(1) # one is not a prime number!!
    False
    """
    "*** YOUR CODE HERE ***"

Hint

Here's a while statement that goes through all numbers above 1 and below n:

i = 2
while i < n:
    ...
    i = i + 1

You can use n % i == 0 to check whether i is a factor of n. If it is, return False.

Description Time: Come up with a one sentence description of the process you implemented to solve is_prime that you think someone could understand without looking at your code. Try not to just read your code, but instead describe the process it carries out.

Recommended, but more challenging (if there's time)

Q3: Repeating

Definition: A positive integer n is a repeating sequence of positive integer m if n is written by repeating the digits of m one or more times. For example, 616161 is a repeating sequence of 61, but 61616 is not.

Implement repeating which takes positive integers t and n. It returns whether n is a repeating sequence of some t-digit integer.

Tip: You can use // and % to separate a positive integer into its last few digits and all the ones before those.

def repeating(t, n):
    """Return whether t digits repeat to form positive integer n.

    >>> repeating(1, 6161)
    False
    >>> repeating(2, 6161)  # repeats 61 (2 digits)
    True
    >>> repeating(3, 6161)
    False
    >>> repeating(4, 6161)  # repeats 6161 (4 digits)
    True
    >>> repeating(5, 6161)  # there are only 4 digits
    False
    """
    if pow(10, t-1) > n:  # make sure n has at least t digits
        return False
    end = _____
    while n:
        if n % pow(10, t) != end:
           return _____
        _____
    return True

Hint

The iterative process needed to implement this function is to repeatedly check that the last t digits of the current n match the last t digits of the original n, then remove the last t digits of the current n.

Q4: Unique Digits

Write a function that returns the number of unique digits in a positive integer. Also implement has_digit, which determines whether a number n contains digit k, and call it in your implementation of unique_digits.

Tip: You can use // 10 and % 10 to separate a positive integer into its one's digit and the rest of its digits.

def unique_digits(n):
    """Return the number of unique digits in positive integer n.

    >>> unique_digits(8675309) # All are unique
    7
    >>> unique_digits(13173131) # 1, 3, and 7
    3
    >>> unique_digits(101) # 0 and 1
    2
    """
    "*** YOUR CODE HERE ***"

def has_digit(n, k):
    """Returns whether k is a digit in n.

    >>> has_digit(10, 1)
    True
    >>> has_digit(12, 7)
    False
    """
    assert k >= 0 and k < 10
    "*** YOUR CODE HERE ***"

Hint

One approach is to loop through every digit from 0 to 9 and check whether n has the digit. Count up the ones it has.