Homework 3: Recursion, Tree Recursion
Due by 11:59pm on Friday, February 17
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 the assignment by uploading all code files you've edited to Gradescope. 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 Gradescope. 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
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.
Q1: Num Eights
Write a recursive function num_eights
that takes a positive integer n
and
returns the number of times the digit 8 appears in n
.
Important: Use recursion; the tests will fail if you use any assignment statements or loops. (You can however use function definitions if you so wish.)
def num_eights(n):
"""Returns the number of times 8 appears as a digit of n.
>>> num_eights(3)
0
>>> num_eights(8)
1
>>> num_eights(88888888)
8
>>> num_eights(2638)
1
>>> num_eights(86380)
2
>>> num_eights(12345)
0
>>> num_eights(8782089)
3
>>> from construct_check import check
>>> # ban all assignment statements
>>> check(HW_SOURCE_FILE, 'num_eights',
... ['Assign', 'AnnAssign', 'AugAssign', 'NamedExpr', 'For', 'While'])
True
"""
"*** YOUR CODE HERE ***"
Use Ok to test your code:
python3 ok -q num_eights
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 8 or contains the digit 8. The first 30 elements of the ping-pong
sequence are listed below, with direction swaps marked using brackets at the
8th, 16th, 18th, 24th, and 28th elements:
Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | [8] | 9 | 10 | 11 | 12 | 13 | 14 | 15 | [16] | 17 | [18] | 19 | 20 | 21 | 22 | 23 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
PingPong Value | 1 | 2 | 3 | 4 | 5 | 6 | 7 | [8] | 7 | 6 | 5 | 4 | 3 | 2 | 1 | [0] | 1 | [2] | 1 | 0 | -1 | -2 | -3 |
Index (cont.) | [24] | 25 | 26 | 27 | [28] | 29 | 30 |
---|---|---|---|---|---|---|---|
PingPong Value | [-4] | -3 | -2 | -1 | [0] | -1 | -2 |
Implement a function pingpong
that returns the nth element of the ping-pong
sequence without using any assignment statements. (You are allowed to use function definitions.)
You may use the function num_eights
, which you defined in the previous question.
Important: Use recursion; the tests will fail if you use any assignment statements. (You can however use function definitions if you so wish.)
Hint: If you're stuck, first try implementing
pingpong
using assignment statements and awhile
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.Hint: There are a few pieces of information that we need to keep track of. One of these details is the direction that we're going (either increasing or decreasing). Building off of the hint above, think about how we can keep track of the direction throughout the calls to the helper function.
def pingpong(n):
"""Return the nth element of the ping-pong sequence.
>>> pingpong(8)
8
>>> pingpong(10)
6
>>> pingpong(15)
1
>>> pingpong(21)
-1
>>> pingpong(22)
-2
>>> pingpong(30)
-2
>>> pingpong(68)
0
>>> pingpong(69)
-1
>>> pingpong(80)
0
>>> pingpong(81)
1
>>> pingpong(82)
0
>>> pingpong(100)
-6
>>> from construct_check import check
>>> # ban assignment statements
>>> check(HW_SOURCE_FILE, 'pingpong',
... ['Assign', 'AnnAssign', 'AugAssign', 'NamedExpr'])
True
"""
"*** YOUR CODE HERE ***"
Use Ok to test your code:
python3 ok -q pingpong
Q3: Count Coins
Given a positive integer change
, a set of coins makes change for change
if
the sum of the values of the coins is change
.
Here we will use standard US Coin values: 1, 5, 10, 25.
For example, the following sets make change for 15
:
- 15 1-cent coins
- 10 1-cent, 1 5-cent coins
- 5 1-cent, 2 5-cent coins
- 5 1-cent, 1 10-cent coins
- 3 5-cent coins
- 1 5-cent, 1 10-cent coin
Thus, there are 6 ways to make change for 15
. Write a recursive function
count_coins
that takes a positive integer change
and returns the number of
ways to make change for change
using coins.
You can use either of the functions given to you:
next_larger_coin
will return the next larger coin denomination from the input, i.e.next_larger_coin(5)
is10
.next_smaller_coin
will return the next smaller coin denomination from the input, i.e.next_smaller_coin(5)
is1
.- Either function will return
None
if the next coin value does not exist
There are two main ways in which you can approach this problem.
One way uses next_larger_coin
, and another uses next_smaller_coin
.
Important: Use recursion; the tests will fail if you use loops.
Hint: Refer the implementation of
count_partitions
for an example of how to count the ways to sum up to a final value with smaller parts. If you need to keep track of more than one value across recursive calls, consider writing a helper function.
def next_larger_coin(coin):
"""Returns the next larger coin in order.
>>> next_larger_coin(1)
5
>>> next_larger_coin(5)
10
>>> next_larger_coin(10)
25
>>> next_larger_coin(2) # Other values return None
"""
if coin == 1:
return 5
elif coin == 5:
return 10
elif coin == 10:
return 25
def next_smaller_coin(coin):
"""Returns the next smaller coin in order.
>>> next_smaller_coin(25)
10
>>> next_smaller_coin(10)
5
>>> next_smaller_coin(5)
1
>>> next_smaller_coin(2) # Other values return None
"""
if coin == 25:
return 10
elif coin == 10:
return 5
elif coin == 5:
return 1
def count_coins(change):
"""Return the number of ways to make change using coins of value of 1, 5, 10, 25.
>>> count_coins(15)
6
>>> count_coins(10)
4
>>> count_coins(20)
9
>>> count_coins(100) # How many ways to make change for a dollar?
242
>>> count_coins(200)
1463
>>> from construct_check import check
>>> # ban iteration
>>> check(HW_SOURCE_FILE, 'count_coins', ['While', 'For'])
True
"""
"*** YOUR CODE HERE ***"
Use Ok to test your code:
python3 ok -q count_coins
Check Your Score Locally
You can locally check your score on each question of this assignment by running
python3 ok --score
This does NOT submit the assignment! When you are satisfied with your score, submit the assignment to Gradescope to receive credit for it.
Submit
Make sure to submit this assignment by uploading any files you've edited to the appropriate Gradescope assignment. For a refresher on how to do this, refer to Lab 00.
Exam Practice
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!
- Fall 2017 MT1 Q4a: Digital
- Summer 2018 MT1 Q5a: Won't You Be My Neighbor?
- Fall 2019 Final Q6b: Palindromes