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Engineering LibreTexts

19.11: Exercises

  • Page ID
    16969
  • Exercise \(\PageIndex{1}\)

    Draw a stack diagram for the following program. What does the program print?

    def b(z):
        prod = a(z, z)
        print z, prod
        return prod
    
    def a(x, y):
        x = x + 1
        return x * y
    
    def c(x, y, z):
        total = x + y + z
        square = b(total)**2
        return square
    
    x = 1
    y = x + 1
    print c(x, y+3, x+y)
    Solution:

    http://thinkpython.com/code/stack_diagram.py.

    Exercise \(\PageIndex{2}\)

    The Ackermann function, A(m, n), is defined:

    \[ A(m,n) = \begin{cases} n+1 & \text{if $m=0$} \\ A(m-1,1) & \text{if $m > 0$ and $n = 0$} \\ A(m-1, A(m,n-1)) & \text{if $m>0$ and $n>0$} \end{cases} \nonumber \]

    See http://en.Wikipedia.org/wiki/Ackermann_function. Write a function named ack that evaluates Ackermann’s function. Use your function to evaluate ack(3, 4), which should be 125. What happens for larger values of m and n?

    Solution:

    http://thinkpython.com/code/ackermann.py.

    Exercise \(\PageIndex{3}\)

    A palindrome is a word that is spelled the same backward and forward, like “noon” and “redivider”. Recursively, a word is a palindrome if the first and last letters are the same and the middle is a palindrome.

    The following are functions that take a string argument and return the first, last, and middle letters:

    def first(word):
        return word[0]
    
    def last(word):
        return word[-1]
    
    def middle(word):
        return word[1:-1]
    

    We’ll see how they work in Chapter 8.

    1. Type these functions into a file named palindrome.py and test them out. What happens if you call middle with a string with two letters? One letter? What about the empty string, which is written '' and contains no letters?
    2. Write a function called is_palindrome that takes a string argument and returns True if it is a palindrome and False otherwise. Remember that you can use the built-in function len to check the length of a string.
    Solution:

    http://thinkpython.com/code/palindrome_soln.py.

    Exercise \(\PageIndex{4}\)

    A number, a, is a power of b if it is divisible by b and a/b is a power of b. Write a function called is_power that takes parameters a and b and returns True if a is a power of b. Note: you will have to think about the base case.

    Exercise \(\PageIndex{5}\)

    The greatest common divisor (GCD) of a and b is the largest number that divides both of them with no remainder.

    One way to find the GCD of two numbers is based on the observation that if r is the remainder when a is divided by b, then gcd(a, b) = gcd(b, r). As a base case, we can use gcd(a, 0) = a.

    Write a function called gcd that takes parameters a and b and returns their greatest common divisor.

    Credit: This exercise is based on an example from Abelson and Sussman’s Structure and Interpretation of Computer Programs.