# Homework 3: Tree Recursion and Data Abstraction hw03.zip

Due by 11:59pm on Monday, 7/9

## Instructions

Download hw03.zip. Inside the archive, you will find a file called hw03.py, along with a copy of the ok autograder.

Submission: When you are done, submit with python3 ok --submit. You may submit more than once before the deadline; only the final submission will be scored. Check that you have successfully submitted your code on okpy.org. See Lab 0 for more instructions on submitting assignments.

Using Ok: If you have any questions about using Ok, please refer to this guide.

Readings: You might find the following references useful:

Grading: Homework is graded based on effort, not correctness. However, there is no partial credit; you must show substantial effort on every problem to receive any points.

The construct_check module is used in this assignment, which defines a function check. For example, a call such as

check("foo.py", "func1", ["While", "For", "Recursion"])

checks that the function func1 in file foo.py does not contain any while or for constructs, and is not an overtly recursive function (i.e., one in which a function contains a call to itself by name.)

## Required questions

### Q1: G function

A mathematical function G on positive integers is defined by two cases:

G(n) = n,                                       if n <= 3
G(n) = G(n - 1) + 2 * G(n - 2) + 3 * G(n - 3),  if n > 3

Write a recursive function g that computes G(n). Then, write an iterative function g_iter that also computes G(n):

Hint: The fibonacci example in the tree recursion lecture is a good illustration of the relationship between the recursive and iterative definitions of a tree recursive problem.

def g(n):
"""Return the value of G(n), computed recursively.

>>> g(1)
1
>>> g(2)
2
>>> g(3)
3
>>> g(4)
10
>>> g(5)
22
>>> from construct_check import check
>>> check(HW_SOURCE_FILE, 'g', ['While', 'For'])
True
"""
"*** YOUR CODE HERE ***"

def g_iter(n):
"""Return the value of G(n), computed iteratively.

>>> g_iter(1)
1
>>> g_iter(2)
2
>>> g_iter(3)
3
>>> g_iter(4)
10
>>> g_iter(5)
22
>>> from construct_check import check
>>> check(HW_SOURCE_FILE, 'g_iter', ['Recursion'])
True
"""
"*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q g
python3 ok -q g_iter

### Q2: Ping-pong

The ping-pong sequence counts up starting from 1 and is always either counting up or counting down. At element k, the direction switches if k is a multiple of 7 or contains the digit 7. The first 30 elements of the ping-pong sequence are listed below, with direction swaps marked using brackets at the 7th, 14th, 17th, 21st, 27th, and 28th elements:

1 2 3 4 5 6 [7] 6 5 4 3 2 1 [0] 1 2 [3] 2 1 0 [-1] 0 1 2 3 4 [5] [4] 5 6

Implement a function pingpong that returns the nth element of the ping-pong sequence without using any assignment statements.

You may use the function has_seven, provided in the homework .py file, which returns True if a number k contains the digit 7 at least once.

Hint: If you're stuck, first try implementing pingpong using assignment statements and a while statement. Then, to convert this into a recursive solution, write a helper function that has a parameter for each variable that changes values in the body of the while loop.

def pingpong(n):
"""Return the nth element of the ping-pong sequence.

>>> pingpong(7)
7
>>> pingpong(8)
6
>>> pingpong(15)
1
>>> pingpong(21)
-1
>>> pingpong(22)
0
>>> pingpong(30)
6
>>> pingpong(68)
2
>>> pingpong(69)
1
>>> pingpong(70)
0
>>> pingpong(71)
1
>>> pingpong(72)
0
>>> pingpong(100)
2
>>> from construct_check import check
>>> check(HW_SOURCE_FILE, 'pingpong', ['Assign', 'AugAssign'])
True
"""
"*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q pingpong

### Q3: Count change

Once the machines take over, the denomination of every coin will be a power of two: 1-cent, 2-cent, 4-cent, 8-cent, 16-cent, etc. There will be no limit to how much a coin can be worth.

Given a positive integer amount, a set of coins makes change for amount if the sum of the values of the coins is amount. For example, the following sets make change for 7:

• 7 1-cent coins
• 5 1-cent, 1 2-cent coins
• 3 1-cent, 2 2-cent coins
• 3 1-cent, 1 4-cent coins
• 1 1-cent, 3 2-cent coins
• 1 1-cent, 1 2-cent, 1 4-cent coins

Thus, there are 6 ways to make change for 7. Write a recursive function count_change that takes a positive integer amount and returns the number of ways to make change for amount using these coins of the future.

Hint: Refer the implementation of count_partitions for an example of how to count the ways to sum up to an amount with smaller parts. If you need to keep track of more than one value across recursive calls, consider writing a helper function.

def count_change(amount):
"""Return the number of ways to make change for amount.

>>> count_change(7)
6
>>> count_change(10)
14
>>> count_change(20)
60
>>> count_change(100)
9828
>>> from construct_check import check
>>> check(HW_SOURCE_FILE, 'count_change', ['While', 'For'])
True
"""
"*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q count_change

### Q4: Towers of Hanoi

A classic puzzle called the Towers of Hanoi is a game that consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with n disks in a neat stack in ascending order of size on a start rod, the smallest at the top, forming a conical shape.

The objective of the puzzle is to move the entire stack to an end rod, obeying the following rules:

• Only one disk may be moved at a time.
• Each move consists of taking the top (smallest) disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
• No disk may be placed on top of a smaller disk.

Complete the definition of move_stack, which prints out the steps required to move n disks from the start rod to the end rod without violating the rules. The provided print_move function will print out the step to move a single disk from the given origin to the given destination.

Hint: Draw out a few games with various n on a piece of paper and try to find a pattern of disk movements that applies to any n. In your solution, take the recursive leap of faith whenever you need to move any amount of disks less than n from one rod to another.

def print_move(origin, destination):
"""Print instructions to move a disk."""
print("Move the top disk from rod", origin, "to rod", destination)

def move_stack(n, start, end):
"""Print the moves required to move n disks on the start pole to the end
pole without violating the rules of Towers of Hanoi.

n -- number of disks
start -- a pole position, either 1, 2, or 3
end -- a pole position, either 1, 2, or 3

There are exactly three poles, and start and end must be different. Assume
that the start pole has at least n disks of increasing size, and the end
pole is either empty or has a top disk larger than the top n start disks.

>>> move_stack(1, 1, 3)
Move the top disk from rod 1 to rod 3
>>> move_stack(2, 1, 3)
Move the top disk from rod 1 to rod 2
Move the top disk from rod 1 to rod 3
Move the top disk from rod 2 to rod 3
>>> move_stack(3, 1, 3)
Move the top disk from rod 1 to rod 3
Move the top disk from rod 1 to rod 2
Move the top disk from rod 3 to rod 2
Move the top disk from rod 1 to rod 3
Move the top disk from rod 2 to rod 1
Move the top disk from rod 2 to rod 3
Move the top disk from rod 1 to rod 3
"""
assert 1 <= start <= 3 and 1 <= end <= 3 and start != end, "Bad start/end"
"*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q move_stack

### Q5: Replace Leaf

Define replace_leaf, which takes a tree t, a value old, and a value new. replace_leaf returns a new tree that's the same as t except that every leaf value equal to old has been replaced with new.

def replace_leaf(t, old, new):
"""Returns a new tree where every leaf value equal to old has
been replaced with new.

>>> yggdrasil = tree('odin',
...                  [tree('balder',
...                        [tree('thor'),
...                         tree('loki')]),
...                   tree('frigg',
...                        [tree('thor')]),
...                   tree('thor',
...                        [tree('sif'),
...                         tree('thor')]),
...                   tree('thor')])
>>> laerad = copy_tree(yggdrasil) # copy yggdrasil for testing purposes
>>> print_tree(replace_leaf(yggdrasil, 'thor', 'freya'))
odin
balder
freya
loki
frigg
freya
thor
sif
freya
freya
>>> laerad == yggdrasil # Make sure original tree is unmodified
True
"""
"*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q replace_leaf

## Extra questions

Extra questions are not worth extra credit and are entirely optional. They are designed to challenge you to think creatively!

### 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.

### Q7: Church numerals

The logician Alonzo Church invented a system of representing non-negative integers entirely using functions. The purpose was to show that functions are sufficient to describe all of number theory: if we have functions, we do not need to assume that numbers exist, but instead we can invent them.

Your goal in this problem is to rediscover this representation known as Church numerals. Here are the definitions of zero, as well as a function that returns one more than its argument:

def zero(f):
return lambda x: x

def successor(n):
return lambda f: lambda x: f(n(f)(x))

First, define functions one and two such that they have the same behavior as successor(zero) and successsor(successor(zero)) respectively, but do not call successor in your implementation.

Next, implement a function church_to_int that converts a church numeral argument to a regular Python integer.

Finally, implement functions add_church, mul_church, and pow_church that perform addition, multiplication, and exponentiation on church numerals.

def one(f):
"""Church numeral 1: same as successor(zero)"""
"*** YOUR CODE HERE ***"

def two(f):
"""Church numeral 2: same as successor(successor(zero))"""
"*** YOUR CODE HERE ***"

three = successor(two)

def church_to_int(n):
"""Convert the Church numeral n to a Python integer.

>>> church_to_int(zero)
0
>>> church_to_int(one)
1
>>> church_to_int(two)
2
>>> church_to_int(three)
3
"""
"*** YOUR CODE HERE ***"

"""Return the Church numeral for m + n, for Church numerals m and n.

5
"""
"*** YOUR CODE HERE ***"

def mul_church(m, n):
"""Return the Church numeral for m * n, for Church numerals m and n.

>>> four = successor(three)
>>> church_to_int(mul_church(two, three))
6
>>> church_to_int(mul_church(three, four))
12
"""
"*** YOUR CODE HERE ***"

def pow_church(m, n):
"""Return the Church numeral m ** n, for Church numerals m and n.

>>> church_to_int(pow_church(two, three))
8
>>> church_to_int(pow_church(three, two))
9
"""
"*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q church_to_int
python3 ok -q add_church
python3 ok -q mul_church
python3 ok -q pow_church