Due by 11:59pm on Thursday, 9/01

## Instructions

Download hw01.zip. The homework problems can be found in the `problems` directory and the quiz problems can be found in the `quiz` directory. You must run `python3 ok --submit` twice: once inside the `problems` directory, and once inside the `quiz` directory.

Submission: When you are done, submit with `python3 ok --submit`. 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 okpy.org. 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:

## Homework Questions

For this set of problems, you must run `ok` from within the `problems` directory. Remember that you may choose to work with a partner on homework questions.

### Question 1

Fill in the blanks in the following function definition 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 = _____
else:
f = _____
return f(a, b)``````

Use OK to test your code:

``python3 ok -q a_plus_abs_b``

### Question 2

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 _____
``````

Use OK to test your code:

``python3 ok -q two_of_three``

### Question 3

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
"""
"*** YOUR CODE HERE ***"
``````

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``

### Question 4

Let's 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` returns the number `1`, but `with_if_function` does not (it can do anything else):

``````def with_if_statement():
"""
>>> with_if_statement()
1
"""
if c():
return t()
else:
return f()

def with_if_function():
return if_function(c(), t(), f())

def c():
"*** YOUR CODE HERE ***"

def t():
"*** YOUR CODE HERE ***"

def f():
"*** YOUR CODE HERE ***"
``````

To test your solution, open an interactive interpreter

``python3 -i hw01.py``

and try calling `with_if_function` and `with_if_statement` to check that one returns 1 and the other doesn't.

Hint: If you are having a hard time identifying how the if statement and if function differ, first try to get them to `print` out different values.

### Question 5

Douglas Hofstadter's Pulitzer-prize-winning book, GĂ¶del, Escher, Bach, poses the following mathematical puzzle.

1. Pick a positive integer `n` as the start.
2. If `n` is even, divide it by 2.
3. If `n` is odd, multiply it by 3 and add 1.
4. 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
"""
"*** YOUR CODE HERE ***"
``````

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``

## Quiz Questions

For this set of problems, you must run `ok` from within the `quiz` directory. While homework questions may be completed with a partner, please remember that quiz questions must be completed alone.

### Question 6: Multiple

Write a function that takes in two numbers and returns the smallest number that is a multiple of both.
``````def multiple(a, b):
"""Return the smallest number n that is a multiple of both a and b.

>>> multiple(3, 4)
12
>>> multiple(14, 21)
42
"""
"*** YOUR CODE HERE ***"
``````

Use OK to test your code:

``python3 ok -q multiple``

### Question 7: Unique Digits

Write a function that returns the number of unique digits in a positive integer.
``````def unique_digits(n):
"""Return the number of unique digits in positive integer n

>>> unique_digits(8675309) # All are unique
7
>>> unique_digits(1313131) # 1 and 3
2
>>> unique_digits(13173131) # 1, 3, and 7
3
>>> unique_digits(10000) # 0 and 1
2
>>> unique_digits(101) # 0 and 1
2
>>> unique_digits(10) # 0 and 1
2
"""
"*** YOUR CODE HERE ***"
``````

Hint: You may find it helpful to first define a function `has_digit(n, k)`, which determines whether a number `n` has digit `k`.

Use OK to test your code:

``python3 ok -q unique_digits``