**Exercise 6.4**. *Draw a stack diagram for the following program. What does the program print? Solution:* http://www.greenteapress.com/thinkpython/code/stack_diagram.py *.*

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)

**Exercise 6.5.** *The Ackermann function, A*(*m*, *n*)*, is de**ﬁ**ned:*

*See* https://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 6.6.** *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* *ﬁ**rst* *and last letters are the same and the middle is a palindrome.*

*The following are functions that take a string argument and return the* *ﬁ**rst**, 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* Strings
*.*

- 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?
- 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 6.7.** *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 6.8.** *The greatest common divisor (GCD) of* *a and* *b is the largest number that divides both of them with no remainder.*

*One way to* *ﬁ**nd* *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*.*

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