Problem 1: Multiple Approaches

Which of the below blocks of code would print something different from the others?

for n in range(10):
    if n % 2 == 0:
        if n <= 10:
            print(n)

for k in range(2,10):
    if not (k % 2 > 0):
        print(k)

j = 0
while j < 10:
    print(j)
    j += 2

for i in range(-2,10,2):
    if i >= 0:
        print(i)

Problem 2: Ranging About

biggest = 0
for i in range(2, 8, 2):
    for j in range(10, -1, -5):
        if i * j > biggest:
            biggest = i * j
print(biggest)

Creating your Algorithms

• Think about how you would solve the problem without a computer. You can’t write code if you don’t understand what you want the computer to do.
• Computers are fast! Brute force methods are often very viable, or at least a good starting point.
• Try to use tools and programming patterns you have already seen. It is often far easier to write programs by assembling pieces from code you have already seen than writing each program entirely from scratch.
  • Identify common patterns that we’ve used to solve past problems
    • Counting things
    • Various ways to loop
    • Tracking biggest/smallest things
    • Checking divisibility

A Recommendation

• One of the most useful things you can do early on is to compile yourself a summary page of various problem-solving approaches
  • Indicate what the problem was, how you solved it, and link to or directly include some sample code
• This makes it far easier in future problems to identify a similar part of a problem and immediately recall (or look back) at how you have solved it previously

Problem 3: Optimus Prime

• Write a function is_prime which takes one integer as input and which returns a boolean that indicates if the input number is a prime number.
• As a reminder, a number is prime if it has no factors besides 1 and itself.
• When you write out your general algorithm, also include information about where you have previously showcased each of the steps you plan to take (what problem, on what date, etc)
  • I may require this sort of information in future homework problems!

Pythagorean Triples

A Pythagorean triplet is a set of three natural numbers, \(a < b < c\), for which, \[ a^2 + b^2 = c^2\]

There exists exactly one Pythagorean triplet where \(a + b + c = 1000\). Find it.

A Solution

# My own attempt below!

def pythagorean(a,b,c):
    """
    Identify the contraints
    1. The Ordering: a < b < c (This prevents duplicate searches, like checking both a=3, b=4 and a=4, b=3).
    2. The Equation: a^2 + b^2 = c^2 (The Pythagorean theorem).
    3. The Sum: a + b + c = 1000.
    
    Reduce the Variables (Mathematical Optimization)
    1. Since a + b + c = 1000, you can rewrite c as: c=1000-a-b
    2. In Pythagorean a^2 + b^2 = (1000 - a - b)^2
    
    Establish Strict Loop Boundaries
    1. Boundaries for a: Since a < b < c, a must be strictly less than a third of the total sum.
       
        Max a < 1000/3 => a <=332
    2. Boundaries for b: b must always start at a + 1 (because a < b). It must also be less than the remaining sum divided by 2.
        Max b < (1000 - a) / 2
    
    ALGORITHMS
    Algorithmic Logic With the math simplified, the programmatic blueprint looks like this:
    1. Initialize a loop for a ranging from 1 to 332.
    2. Initialize a nested loop for b ranging from (a + 1) to (1000 - a) / 2.
    3. Calculate c inside the loop: c = 1000 - a - b.
    4. Test the condition: Check if a^2 + b^2 == c^2.
        Return the result: If the condition is met,
        multiply a x b x c to get the final answer and break the loop immediately (since the problem states there is exactly one solution).
    
    """
    for a in range(1,333):
        for b in range(a+1, int((1000 - a)/2)):
            c = 1000 - a - b
            if a**2 + b**2 == c**2:
                return a * b * c 
            
            
            
print(pythagorean(10 ,40 , 55))