# Homework 3 Solutions

**Solutions:** You can find the file with solutions for all
questions here.

## Required questions

### Question 1: Falling Factorial

Let's write a function `falling`

, which is a "falling" factorial
that takes two arguments, `n`

and `k`

, and returns the product of `k`

consecutive numbers, starting from `n`

and working downwards.

If `k`

is larger than n, only multiply up to n consecutive numbers!

```
def falling(n, k):
"""Compute the falling factorial of n to depth k.
>>> falling(6, 3) # 6 * 5 * 4
120
>>> falling(4, 0)
1
>>> falling(4, 3) # 4 * 3 * 2
24
>>> falling(4, 1) # 4
4
>>> falling(4, 10) # 4 * 3 * 2 * 1 # Only n times!!
24
"""
if k > n:
k = n
total, stop = 1, n-k
while n > stop:
total, n = total*n, n-1
return total
```

Use OK to test your code:

`python3 ok -q falling --local`

### Question 2: Nonzero

Write a function that takes in a list and returns the first nonzero entry.

```
def nonzero(lst):
""" Returns the first nonzero element of a list
>>> nonzero([1, 2, 3])
1
>>> nonzero([0, 1, 2])
1
>>> nonzero([0, 0, 0, 0, 0, 0, 5, 0, 6])
5
"""
for i in lst:
if i != 0:
return i
```

Use OK to test your code:

`python3 ok -q nonzero --local`

### Question 3: Hailstone

Complete this question using iteration!

Douglas Hofstadter's Pulitzer-prize-winning book, *GĂ¶del, 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 sequence of values of `n`

is often called a Hailstone sequence,
because hailstones also travel up and down in the atmosphere before
falling to earth. 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
"""
s = 1
print(n)
while n != 1:
if n % 2 == 0:
n = n // 2
else:
n = n * 3 + 1
print(n)
s = s + 1
return s
```

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

Use OK to test your code:

`python3 ok -q hailstone --local`

### Question 4: Classify the elements

Complete the function `odd_even`

that classifies an number as either `'odd'`

or `'even'`

and the function `classify`

that takes in a list and applies `odd_even`

to all elements in the list.

```
def odd_even(x):
"""
Classify a number as odd or even.
>>> odd_even(4)
'even'
>>> odd_even(3)
'odd'
"""
if (x % 2) == 0:
return 'even'
else:
return 'odd'
def classify(s):
"""
Classify all the elements of a sequence as odd or even
>>> classify([0, 1, 2, 4])
['even', 'odd', 'even', 'even']
"""
return [odd_even(x) for x in s]
```

Use OK to test your code:

`python3 ok -q odd_even --local`

Use OK to test your code:

`python3 ok -q classify --local`

### Question 5: Decode

Implement a function `decode`

, which takes in a list of pairs of numbers and returns a list of lists of decoded values.

- The list contains pairs of the form
`[sex, age]`

- Sex is an int that is either 0 or 1 and age is an int between 0 and 10
- Return a list of strings where the Sex gets replaced by "Male" for 0 and "Female" for 1, and age gets replaced by "0-9", "10-19", ..., "90-99", "100+"

See the doctests for examples.

One other thing: your answer can only be *one line long*. You should
make use of list comprehensions and use the helper function!

```
def decode_helper(pair):
"""
Optional helper function! Could be useful to turn something like [0, 0] to 'Male 0-9'
"""
result = ""
if pair[0] == 0:
result += "Male "
else:
result += "Female "
if pair[1] == 0:
result += '0-9'
elif pair[1] == 10:
result += '100+'
else:
result += str(pair[1]) + '0-' + str(pair[1]) + '9'
return result
def decode(list_of_sex_age_pairs):
"""
>>> decode([[0, 0], [1, 1], [1, 10]])
['Male 0-9', 'Female 10-19', 'Female 100+']
>>> decode([[0, 0], [0, 1], [0, 2], [0, 3], [0, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10]])
['Male 0-9', 'Male 10-19', 'Male 20-29', 'Male 30-39', 'Male 40-49', 'Female 50-59', 'Female 60-69', 'Female 70-79', 'Female 80-89', 'Female 90-99', 'Female 100+']
"""
return [decode_helper(pair) for pair in list_of_sex_age_pairs]
```

Use OK to test your code:

`python3 ok -q decode --local`