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 (since 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).
    
    """
    # Now Write the program 

Prep

• All problems today revolve around troubleshooting and finding errors in some “launch code”-esque checks
• You will want at least someone or all of you to have downloaded all the lecture contents linked in Discord
• To get rich installed easily, I have included a Python file in the contents that you should just need to run, and it will (in theory) handle things automatically for you.
  • You’ll need this for the problems today

Problem 1: Stress Tests

• The p1.py file is doing some stress testing, but has some issues
• Focus on using print statements to track down where things are going wrong
• How could you get around this issue?

Problem 2: Systems Go?

• The p2.py file should be saying that all systems are a go, but it is not
• Pay particular attention to the documentation to figure out what has gone wrong
• You should be able to fix all the problems by only changing 3-5 lines

Problem 3: Sufficient Shields?

• The p3.py file should also be saying that all systems are go, but is not
• Instead of using print statements here, use only your debugger to track down what is happening and how to fix it

Problem 4: System Scanning

• Finally, p4.py should be verifying a scan, but is instead erring out. Use the printed error alongside print statements or the debugger to establish where things have gone awry
• How can you easily fix this problem?