# Homework 1 Solutions

## Solution Files

You can find the solutions in hw01.py.

## Homework Questions

### Q0: Welcome Survey

Please complete this welcome survey before you submit your homework. Your email address will be collected to verify that you completed the survey and to correlate responses across surveys, but your individual responses will only be read after responses are anonymized.

### Q1: A Plus Abs B

Fill in the blanks in the following function for adding `a`

to the
absolute value of `b`

, without calling `abs`

.

```
from operator import add, sub
def a_plus_abs_b(a, b):
"""Return a+abs(b), but without calling abs.
>>> a_plus_abs_b(2, 3)
5
>>> a_plus_abs_b(2, -3)
5
"""
if b < 0:
f = sub else:
f = add return f(a, b)
```

Use Ok to test your code:

`python3 ok -q a_plus_abs_b`

If `b`

is positive, we add the numbers together. If `b`

is negative, we
subtract the numbers. Therefore, we choose the operator `add`

or `sub`

based on the sign of `b`

.

Video walkthrough: https://youtu.be/o9eUNrWTr3I

### Q2: Two of Three

Write a function that takes three *positive* numbers and returns the sum
of the squares of the two largest numbers. **Use only a single line for
the body of the function.**

```
def two_of_three(a, b, c):
"""Return x*x + y*y, where x and y are the two largest members of the
positive numbers a, b, and c.
>>> two_of_three(1, 2, 3)
13
>>> two_of_three(5, 3, 1)
34
>>> two_of_three(10, 2, 8)
164
>>> two_of_three(5, 5, 5)
50
"""
return max(a*a+b*b, a*a+c*c, b*b+c*c)
# Alternate solution
return a**2 + b**2 + c**2 - min(a, b, c)**2
```

Hint:Consider using the`max`

or`min`

function:`>>> max(1, 2, 3) 3 >>> min(-1, -2, -3) -3`

Use Ok to test your code:

`python3 ok -q two_of_three`

We use the fact that if `a>b`

and `b>0`

, then `square(a)>square(b)`

.
So, we can take the `max`

of the sum of squares of all pairs. The
`max`

function can take an arbitrary number of arguments.

Alternatively, we can do the sum of squares of all the numbers. Then we pick the smallest value, and subtract the square of that.

Video walkthrough: https://youtu.be/oPN3OCGGb4M

### Q3: Largest Factor

Write a function that takes an integer `n`

that is **greater than 1** and
returns the largest integer that is smaller than `n`

and evenly divides `n`

.

```
def largest_factor(n):
"""Return the largest factor of n that is smaller than n.
>>> largest_factor(15) # factors are 1, 3, 5
5
>>> largest_factor(80) # factors are 1, 2, 4, 5, 8, 10, 16, 20, 40
40
>>> largest_factor(13) # factor is 1 since 13 is prime
1
"""
factor = n - 1
while factor > 0:
if n % factor == 0:
return factor
factor -= 1
```

Hint:To check if`b`

evenly divides`a`

, you can use the expression`a % b == 0`

, which can be read as, "the remainder of dividing`a`

by`b`

is 0."

Use Ok to test your code:

`python3 ok -q largest_factor`

Iterating from `n-1`

to 1, we return the first integer that evenly divides
`n`

. This is guaranteed to be the largest factor of `n`

.

Video walkthrough: https://youtu.be/pVgxbeL4DHQ

### Q4: If Function vs Statement

Let's try to write a function that does the same thing as an `if`

statement.

```
def if_function(condition, true_result, false_result):
"""Return true_result if condition is a true value, and
false_result otherwise.
>>> if_function(True, 2, 3)
2
>>> if_function(False, 2, 3)
3
>>> if_function(3==2, 3+2, 3-2)
1
>>> if_function(3>2, 3+2, 3-2)
5
"""
if condition:
return true_result
else:
return false_result
```

Despite the doctests above, this function actually does *not* do the
same thing as an `if`

statement in all cases. To prove this fact,
write functions `c`

, `t`

, and `f`

such that `with_if_statement`

prints the number `2`

, but `with_if_function`

prints both `1`

and `2`

.

```
def with_if_statement():
"""
>>> result = with_if_statement()
2
>>> print(result)
None
"""
if c():
return t()
else:
return f()
def with_if_function():
"""
>>> result = with_if_function()
1
2
>>> print(result)
None
"""
return if_function(c(), t(), f())
def c():
return False
def t():
print(1)
def f():
print(2)
```

Hint: If you are having a hard time identifying how an`if`

statement and`if_function`

differ, consider the rules of evaluation for`if`

statements and call expressions.

Use Ok to test your code:

```
python3 ok -q with_if_statement
python3 ok -q with_if_function
```

The function `with_if_function`

uses a call expression, which
guarantees that all of its operand subexpressions will be evaluated
before `if_function`

is applied to the resulting arguments.

Therefore, even if `c`

returns `False`

, the function `t`

will be called. When
we call `t`

, we print out `1`

. Then, when we call `f`

, we will also print `2`

.

By contrast, `with_if_statement`

will never call `t`

if `c`

returns
`False`

. Thus, we will only call `f`

, printing `2`

.

### Q5: Hailstone

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 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
"""
length = 1
while n != 1:
print(n)
if n % 2 == 0:
n = n // 2 # Integer division prevents "1.0" output
else:
n = 3 * n + 1
length = length + 1
print(n) # n is now 1
return length
```

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`

We keep track of the current length of the hailstone sequence and the current value of the hailstone sequence. From there, we loop until we hit the end of the sequence, updating the length in each step.

Note: we need to do floor division `//`

to remove decimals.

Video walkthrough: https://youtu.be/lZZQ0BpsXIc

## Extra questions

Extra questions are not worth extra credit and are entirely optional. They are designed to challenge you to think creatively! They do not resemble any exam questions or relate directly to any required content in the course. Feel free to skip them.

### Q6: Quine

Write a one-line program that prints itself, using only the following features of the Python language:

- Number literals
- Assignment statements
- String literals that can be expressed using single or double quotes
- The arithmetic operators
`+`

,`-`

,`*`

, and`/`

- The built-in
`print`

function - The built-in
`eval`

function, which evaluates a string as a Python expression - The built-in
`repr`

function, which returns an expression that evaluates to its argument

You can concatenate two strings by adding them together with `+`

and repeat a
string by multipying it by an integer. Semicolons can be used to separate
multiple statements on the same line. E.g.,

```
>>> c='c';print('a');print('b' + c * 2)
a
bcc
```

Hint: Explore the relationship between single quotes, double quotes, and the`repr`

function applied to strings.

A program that prints itself is called a Quine. Place your solution in the multi-line string named `quine`

.

*Note*: No tests will be run on your solution to this problem.