# Homework 2: Higher Order Functions

*Due by 11:59pm on Thursday, February 3*

## Instructions

Download hw02.zip. Inside the archive, you will find
a file called hw02.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
correctness. Each incorrect problem will decrease the total score by one point. There is a homework recovery policy as stated in the syllabus.
**This homework is out of 2 points.**

# Required questions

Several doctests refer to these functions:

```
from operator import add, mul
square = lambda x: x * x
identity = lambda x: x
triple = lambda x: 3 * x
increment = lambda x: x + 1
```

## Getting Started Videos

These videos may provide some helpful direction for tackling the coding problems on this assignment.

To see these videos, you should be logged into your berkeley.edu email.

## Parsons Problems

To work on these problems, open the Parsons editor:

`python3 parsons`

### Q1: Count Until Larger

Implement the function `count_until_larger`

. `count_until_larger`

takes in a positive integer `num`

. `count_until_larger`

counts the distance between the rightmost digit of `num`

and the nearest greater digit; to do so, the function counts digits from right to left. Once it encounters a digit larger than the rightmost digit, it returns that count. If no such digit exists, then the function returns `-1`

.

For example, `8117`

has a rightmost digit of `7`

and returns a count of `3`

. `9118117`

also returns a count of `3`

: for both, the count stops at `8`

.

`0`

should be treated as having no digits and returns a count of `-1`

.

Consult the following doctests for specific behaviors of `count_until_larger`

.

```
def count_until_larger(num):
"""
Complete the function count_until_larger that takes in a positive integer num.
count_until_larger examines the rightmost digit and counts digits from right to
left until it encounters a digit larger than the rightmost digit, then returns that count.
>>> count_until_larger(117) # .Case 1
-1
>>> count_until_larger(8117) # .Case 2
3
>>> count_until_larger(9118117) # .Case 3
3
>>> count_until_larger(8777) # .Case 4
3
>>> count_until_larger(22) # .Case 5
-1
>>> count_until_larger(0) # .Case 6
-1
"""
"*** YOUR CODE HERE ***"
```

### Q2: Filter Sequence

Write a function `filter_sequence`

which takes in two integers, `start`

and `stop`

, as well as a function `cond`

, which takes in a single argument and outputs a boolean value. `filter_sequence`

returns the sum of all digits from `start`

to `stop`

(inclusive) for which `cond`

returns `True`

.

```
def filter_sequence(cond, start, stop):
"""
Returns the sum of numbers from start (inclusive) to stop (inclusive) that satisfy
the one-argument function cond.
>>> filter_sequence(lambda x: x % 2 == 0, 0, 10) # .Case 1
30
>>> filter_sequence(lambda x: x % 2 == 1, 0, 10) # .Case 2
25
"""
"*** YOUR CODE HERE ***"
```

## Code Writing Questions

### Q3: Hailstone

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

- Pick a positive integer
`n`

as the start. - If
`n`

is even, divide it by 2. - If
`n`

is odd, multiply it by 3 and add 1. - 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
>>> b = hailstone(1)
1
>>> b
1
"""
"*** YOUR CODE HERE ***"
```

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

Note that if

`n == 1`

initially, then the sequence is one step long.

Use Ok to test your code:

`python3 ok -q hailstone`

**Curious about hailstones or hailstone sequences? Take a look at these articles:**

- Check out this article to learn more about how hailstones work!
- In 2019, there was a major development in understanding how the hailstone conjecture works for most numbers!

### Q4: Product

The `summation(n, term)`

function from the higher-order functions lecture adds
up `term(1) + ... + term(n)`

. Write a similar function called `product`

that
returns `term(1) * ... * term(n)`

.

```
def product(n, term):
"""Return the product of the first n terms in a sequence.
n: a positive integer
term: a function that takes one argument to produce the term
>>> product(3, identity) # 1 * 2 * 3
6
>>> product(5, identity) # 1 * 2 * 3 * 4 * 5
120
>>> product(3, square) # 1^2 * 2^2 * 3^2
36
>>> product(5, square) # 1^2 * 2^2 * 3^2 * 4^2 * 5^2
14400
>>> product(3, increment) # (1+1) * (2+1) * (3+1)
24
>>> product(3, triple) # 1*3 * 2*3 * 3*3
162
"""
"*** YOUR CODE HERE ***"
```

Use Ok to test your code:

`python3 ok -q product`

### Q5: Accumulate

Let's take a look at how `summation`

and `product`

are instances of a more
general function called `accumulate`

, which we would like to implement:

```
def accumulate(merger, start, n, term):
"""Return the result of merging the first n terms in a sequence and start.
The terms to be merged are term(1), term(2), ..., term(n). merger is a
two-argument commutative function.
>>> accumulate(add, 0, 5, identity) # 0 + 1 + 2 + 3 + 4 + 5
15
>>> accumulate(add, 11, 5, identity) # 11 + 1 + 2 + 3 + 4 + 5
26
>>> accumulate(add, 11, 0, identity) # 11
11
>>> accumulate(add, 11, 3, square) # 11 + 1^2 + 2^2 + 3^2
25
>>> accumulate(mul, 2, 3, square) # 2 * 1^2 * 2^2 * 3^2
72
>>> # 2 + (1^2 + 1) + (2^2 + 1) + (3^2 + 1)
>>> accumulate(lambda x, y: x + y + 1, 2, 3, square)
19
>>> # ((2 * 1^2 * 2) * 2^2 * 2) * 3^2 * 2
>>> accumulate(lambda x, y: 2 * x * y, 2, 3, square)
576
>>> accumulate(lambda x, y: (x + y) % 17, 19, 20, square)
16
"""
"*** YOUR CODE HERE ***"
```

`accumulate`

has the following parameters:

`term`

and`n`

: the same parameters as in`summation`

and`product`

`merger`

: a two-argument function that specifies how the current term is merged with the previously accumulated terms.`start`

: value at which to start the accumulation.

For example, the result of `accumulate(add, 11, 3, square)`

is

`11 + square(1) + square(2) + square(3) = 25`

Note:You may assume that`merger`

is commutative. That is,`merger(a, b) == merger(b, a)`

for all`a`

,`b`

, and`c`

. However, you may not assume`merger`

is chosen from a fixed function set and hard-code the solution.

After implementing `accumulate`

, show how `summation`

and `product`

can both be
defined as function calls to `accumulate`

.

**Important:**
You should have a single line of code (which should be a `return`

statement)
in each of your implementations for `summation_using_accumulate`

and
`product_using_accumulate`

, which the syntax check will check for.

Use Ok to test your code:

```
python3 ok -q accumulate
python3 ok -q summation_using_accumulate
python3 ok -q product_using_accumulate
```

## Submit

Make sure to submit this assignment by running:

`python3 ok --submit`

## Bonus Questions

Homework assignments will also contain prior exam-level questions for you to take a look at. These questions have no submission component; feel free to attempt them if you'd like a challenge!

Note that exams from Spring 2020, Fall 2020, and Spring 2021 gave students access to an interpreter, so the question format may be different than other years. Regardless, the questions included remain good exam-level problems doable

withoutaccess to an interpreter.

- Fall 2019 MT1 Q3: You Again [Higher Order Functions]
- Spring 2021 MT1 Q4: Domain on the Range [Higher Order Functions]
- Fall 2021 MT1 Q1b: tik [Functions and Expressions]