Homework 1 Solutions
Review from lab
Division
Let's compare the different divisionrelated operators in Python:
True Division (decimal division) The / Operator 
Floor Division (integer division) The // Operator 
Modulo (similar to a remainder) The % Operator 




Note that floor division and modulo both return an integer while true division always returns a floating point number.
For now, the only difference you need to understand between integers and floats is that
integers are positive or negative whole numbers, while floats are positive or negative
decimals.
One useful technique involving the %
operator is to check
whether a number x
is divisible by another number y
:
x % y == 0
For example, in order to check if x
is an even number:
x % 2 == 0
Floats and precedence
Floating point numbers (floats) behave a lot like real numbers. You can identify a float by the decimal point. All floats have decimal points. To write a floating point number (as a literal), you must add a decimal point!
>>> 3.141592
3.141592
>>> 2 * 3.141592 # you can mix ints and floats
6.283184
>>> pie = 3.141592 # you can assign values to variables
>>> pie
3.141592
>>> pie / pie
1.0
>>> pie / pie == 1 # a float can be equal in value to an int
True
>>> from math import pi # here is a better pi
>>> pi
3.141592653589793
>>> 5.0 / 3.0 # this is division of floats, not ints
1.6666666666666667
>>> 2 ** (1 / 2) # square root  isn't that transcendental?
1.4142135623730951
Expressions follow operator precedence (just like in math). Operations are performed one at a time in a specific order. Parenthesis are used to specify order (again, just like in math  remember PEMDAS?).
>>> 2 + 3  4 + 5 # equal precedence, left to right
6
>>> 2 + 3  (4 + 5) # order matters  parentheses are your friend
4
>>> (((2 + 3)  4) + 5) # explicit order of the first example
6
>>> 2 + 3 * 4 # * and / bind more tightly than + or 
14
>>> 2 + (3 * 4)
14
>>> (2 + 3) * 4
20
>>> 2 + 3 / 4 * 5 # what about * and / ?
5.75
An expression can have multiple return values, called a tuple:
>>> 2, 3
(2, 3)
>>> x, y = 1, pi
>>> x
1
>>> y
3.141592653589793
Strings
It is very useful to be able to write programs that operate on strings, not just numbers. Without strings, web browsers and word processors would be like the matrix! Just as with ints, floats, and booleans, strings are a data type and have certain operators defined on them:
>>> 'cal' # a string literal is a sequence of characters in quotes
'cal'
>>> "rocks" # either kind of quote, but they need to match
'rocks'
>>> "cal" + "rocks" # + is concatenation
'calrocks'
>>> 'cal' * 3 # * is replication
'calcalcal'
>>> 'cal' == "cal" # equality is if they are the same string (notice the single and double quotes)
True
>>> 'Cal' == 'cal' # case sensitive
False
>>> 'Cal' < 'cal' # lexicographic ordering, with upper before lower case
True
>>> 'you' is not 'me'
True
Problems
Question 1: Oddly we go
Define odd
, which takes an integer and returns whether it is odd.
Your solution will look like return <expression>
.
def odd(number):
"""Return whether the number is odd.
>>> odd(2)
False
>>> odd(5)
True
"""
return number % 2 == 1
Use OK to test your code:
python3 ok q odd
Question 2: Distance
Implement a function called distance(x1, y1, x2, y2)
where:
x1
andy1
form an xy coordinate pairx2
andy2
form an xy coordinate pair
distance
returns the Euclidean distance between the two points. Use the
following formula:
from math import sqrt
def distance(x1, y1, x2, y2):
"""Calculates the Euclidian distance between two points (x1, y1) and (x2, y2)
>>> distance(1, 1, 1, 2)
1.0
>>> distance(1, 3, 1, 1)
2.0
>>> distance(1, 2, 3, 4)
2.8284271247461903
"""
return sqrt((y2y1)**2 + (x2x1)**2)
Use OK to test your code:
python3 ok q distance
Question 3: Distance (3D)
Now, let us edit this program to get the distance between two
3dimensional coordinates. Your distance3d
function should take six
arguments and compute the following:
def distance3d(x1, y1, z1, x2, y2, z2):
"""Calculates the 3D Euclidian distance between two points (x1, y1, z1) and
(x2, y2, z2).
>>> distance3d(1, 1, 1, 1, 2, 1)
1.0
>>> distance3d(2, 3, 5, 5, 8, 3)
6.164414002968976
"""
return sqrt((y2y1)**2 + (x2x1)**2 + (z2z1)**2)
Use OK to test your code:
python3 ok q distance3d
Question 4: Add absolute
Implement the function a_plus_abs_b
which takes 2 numbers a
and b
,
and returns the result of doing a
plus the absolute value of b
.
You must not modify the template in the starter code, use the abs
function, or define any new functions  you can only fill in the blanks.
Hint: We've seen that we can name a value by assigning it to a variable. Functions are values too. Try typing the name of a function you have defined or imported into the python interpreter, and then assigning it to a variable, e.g.:
>>> from operator import add
>>> my_func = add
>>> my_func(5, 10) == add(5, 10)
True
>>> type(my_func)
<class 'builtin_function_or_method'>
>> type(add)
<class 'builtin_function_or_method'>
How can you use the imported functions to solve this problem?
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
>>> a_plus_abs_b(5, 1)
4
"""
if b < 0:
f = sub
else:
f = add
return f(a, b) # You can replace this line, but don't have to.
Use OK to test your code:
python3 ok q a_plus_abs_b
We choose the operator add
or sub
based on the sign of b
.
Question 5: Quadratic Formula
Complete the function so that it returns a tuple
that contains both roots of a quadratic polynomial using the quadratic formula. The first element in the tuple should be the "plus" case and the second element should be the "minus" case. Your solution must call the sqrt
function
exactly once.
See here for a basic tuple example.
from math import sqrt
def quadratic(a, b, c):
"""
>>> quadratic(1, 0, 1)
(1.0, 1.0)
>>> quadratic(1, 2, 1)
(1.0, 1.0)
>>> quadratic(1, 3, 4)
(1.0, 4.0)
"""
t = sqrt(b*b  4*a*c)
return (b+t)/(2*a),(bt)/(2*a)
Use OK to test your code:
python3 ok q quadratic
We use a temporary variable to hold the subexpression within the sqrt.
Question 6: 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, 3) # 4 * 3 * 2
24
>>> falling(4, 1) # 4
4
>>> falling(4, 0)
1
"""
total = 1
if k > n:
k = n
while k > 0:
total *= n
n = 1
k = 1
return total
Use OK to test your code:
python3 ok q falling
Question 7: Hailstone
Complete this question using iteration!
Douglas Hofstadter's Pulitzerprizewinning 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:
Note: Use floor division (can you think of why?)
def hailstone(n):
"""Print the hailstone sequence starting at n and return its
length.
>>> a = hailstone(2)
2
1
>>> a
2
>>> b = hailstone(10)
10
5
16
8
4
2
1
>>> b
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
Optional Questions
Question 8: Harmonic Mean
Implement harmonic_mean
, which returns the harmonic mean of two positive numbers
x
and y
. The harmonic mean of 2 numbers is 2 divided by the sum of the
reciprocals of the numbers. (The reciprocal of x
is 1/x
.)
def harmonic_mean(x, y):
"""Return the harmonic mean of x and y.
>>> harmonic_mean(2, 6)
3.0
>>> harmonic_mean(1, 1)
1.0
>>> harmonic_mean(2.5, 7.5)
3.75
>>> harmonic_mean(4, 12)
6.0
"""
return 2/(1/x+1/y)
Use OK to test your code:
python3 ok q harmonic_mean
Question 9: 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 expression 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 (a*a + b*b + c*c)  min(a*a, b*b, c*c)
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.