Python/project_euler/problem_009/sol2.py

"""
Problem 9: https://projecteuler.net/problem=9

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
    a^2 + b^2 = c^2
For example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2.

There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.
"""


def solution(n: int = 1000) -> int:
    """
     Return the product of a,b,c which are Pythagorean Triplet that satisfies
     the following:
     1. a < b < c
     2. a**2 + b**2 = c**2
     3. a + b + c = n

    >>> solution(1000)
    31875000
    """
    product = -1
    candidate = 0
    for a in range(1, n // 3):
        """Solving the two equations a**2+b**2=c**2 and a+b+c=N eliminating c"""
        b = (n * n - 2 * a * n) // (2 * n - 2 * a)
        c = n - a - b
        if c * c == (a * a + b * b):
            candidate = a * b * c
            if candidate >= product:
                product = candidate
    return product


if __name__ == "__main__":
    print(solution(int(input().strip())))