Due by 11:59pm on Wednesday, October 15

Instructions

Download hw06.zip. Inside the archive, you will find a file called hw06.py, along with a copy of the ok autograder.

Submission: When you are done, submit the assignment 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:

• Section 1.7.4

Grading: Homework is graded based on correctness. Each incorrect problem will decrease the total score by one point. This homework is out of 6 points.

Required Questions

Q1: Max Product

Implement max_product, which takes a list of numbers and returns the maximum product that can be formed by multiplying together non-consecutive elements of the list. Assume that all numbers in the input list are greater than or equal to 1.

def max_product(s):
    """Return the maximum product of non-consecutive elements of s.

    >>> max_product([10, 3, 1, 9, 2])   # 10 * 9
    90
    >>> max_product([5, 10, 5, 10, 5])  # 5 * 5 * 5
    125
    >>> max_product([])                 # The product of no numbers is 1
    1
    """
    "*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q max_product
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Q2: Count Coins

Given a positive integer total, a set of coins makes change for total if the sum of the values of the coins is total. 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 total and returns the number of ways to make change for total 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) is 10.
• next_smaller_coin will return the next smaller coin denomination from the input, i.e. next_smaller_coin(5) is 1.
• 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. It is up to you which one you want to use!

Important: Use recursion; the tests will fail if you use loops.

Hint: Refer to 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(total):
    """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(SOURCE_FILE, 'count_coins', ['While', 'For'])
    True
    """
    "*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q count_coins
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Q3: Super Mario

Mario needs to jump over a sequence of Piranha plants called level. level is represented as a string of empty spaces (' '), indicating no plant, and P's ('P'), indicating the presence of a plant. Mario only moves forward and can either step (move forward one spot) or jump (move forward two spots) from each position. How many different ways can Mario traverse a level without stepping or jumping into a Piranha plant ('P')? Assume that every level begins with an empty space (' '), where Mario starts, and ends with an empty space (' '), where Mario must end up.

Hint: You can access the ith character in a string s by using s[i]. For example:

>>> s = 'abcdefg'
>>> s[0]
'a'
>>> s[2]
'c'

Hint: You can find the total number of characters in a string using the built-in len function:

>>> s = 'abcdefg'
>>> len(s)
7
>>> len('')
0

def mario_number(level):
    """Return the number of ways that Mario can perform a sequence of steps
    or jumps to reach the end of the level without ever landing in a Piranha
    plant. Assume that every level begins and ends with a space.

    >>> mario_number(' P P ')   # jump, jump
    1
    >>> mario_number(' P P  ')   # jump, jump, step
    1
    >>> mario_number('  P P ')  # step, jump, jump
    1
    >>> mario_number('   P P ') # step, step, jump, jump or jump, jump, jump
    2
    >>> mario_number(' P PP ')  # Mario cannot jump two plants
    0
    >>> mario_number('    ')    # step, jump ; jump, step ; step, step, step
    3
    >>> mario_number('    P    ')
    9
    >>> mario_number('   P    P P   P  P P    P     P ')
    180
    """
    "*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q mario_number
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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 Assignment

Submit this assignment by uploading any files you've edited to the appropriate Gradescope assignment. Lab 00 has detailed instructions.