## Starter Files

Download lab06.zip. Inside the archive, you will find starter files for the questions in this lab, along with a copy of the OK autograder.

## Recursion

### Question 1: AB Plus C

Implement `ab_plus_c`, a function that takes arguments `a`, `b`, and `c` and computes `a * b + c`. You can assume a and b are both nonnegative integers. However, you can't use the `*` operator. Try recursion!

``````def ab_plus_c(a, b, c):
"""Computes a * b + c.

>>> ab_plus_c(2, 4, 3)  # 2 * 4 + 3
11
>>> ab_plus_c(0, 3, 2)  # 0 * 3 + 2
2
>>> ab_plus_c(3, 0, 2)  # 3 * 0 + 2
2
"""
if b == 0:
return c
return a + ab_plus_c(a, b - 1, c)``````

Use OK to test your code:

``python3 ok -q ab_plus_c``

### Question 2: Filter

Write the recursive version of the function `filter` which returns a list and takes in

• `f` - a one-argument function that returns `True` if the passed in argument should be included in the resulting list or `False` otherwise
• `seq` - a list of values

Note that this is different from the built in `filter` function we learned previously, which returns a filter object, not a list.

``````def filter(f, seq):
"""Filter a sequence to only contain values allowed by filter.

>>> def is_even(x):
...     return x % 2 == 0
>>> def divisible_by5(x):
...     return x % 5 == 0
>>> filter(is_even, [1,2,3,4])
[2, 4]
>>> filter(divisible_by5, [1, 4, 9, 16, 25, 100])
[25, 100]
>>> filter(is_even, [1])
[]
>>> filter(is_even, [2])
[2]
>>> filter(is_even, [])
[]
"""

if seq == []:
return seq
if f(seq[0]):
return [seq[0]] + filter(f, seq[1:])
return filter(f, seq[1:])``````

Use OK to test your code:

``python3 ok -q filter``

### Question 3: Decimal

Write the recursive version of the function `decimal` which takes in an integer `n` and returns a list of its digits, the decimal representation of `n`. See the doctests to handle the case where `n < 0`.

``````def decimal(n):
"""Return a list representing the decimal representation of a number.

>>> decimal(0)
[0]
>>> decimal(2)
[2]
>>> decimal(-8)
['-', 8]
>>> decimal(-136)
['-', 1, 3, 6]
>>> decimal(55055)
[5, 5, 0, 5, 5]
"""
if n < 0:
return ['-'] + decimal(-1 * n)
elif n < 10:
return [n]
else:
return decimal(n // 10) + [n % 10]``````

Use OK to test your code:

``python3 ok -q decimal``

### Question 4: Insect Combinatorics

Consider an insect in an M by N grid. The insect starts at the bottom left corner, (0, 0), and wants to end up at the top right corner, (M-1, N-1). The insect is only capable of moving right or up. Write a function `paths` that takes a grid length and width and returns the number of different paths the insect can take from the start to the goal. (There is a closed-form solution to this problem, but try to answer it procedurally using recursion.)

For example, the 2 by 2 grid has a total of two ways for the insect to move from the start to the goal. For the 3 by 3 grid, the insect has 6 different paths (only 3 are shown above). Note that this problem uses tree recursion.

``````def paths(m, n):
"""Return the number of paths from one corner of an
M by N grid to the opposite corner.

>>> paths(2, 2)
2
>>> paths(5, 7)
210
>>> paths(117, 1)
1
>>> paths(1, 157)
1
"""
if m == 0 or n == 0:
return 0
if m == 1 and n == 1:
return 1
return paths(m - 1, n) + paths(m, n - 1)``````

Use OK to test your code:

``python3 ok -q paths``

## Submission

When you are done, submit your file to Gradescope. You only need to upload the following files:

• `lab06.py`