# Euler Problem 9¶

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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 $abc$.

We start bruteforcing even though it feels like there may be a smart solution.

In [1]:

    import timeit

    def brute_force():
        for a in range(1, 251):
            for b in range(a + 1, 501):
                for c in range(b + 1, 1001):
                    if a + b + c == 1000 and a * a + b * b == c * c:
                        return a * b * c

    print(timeit.timeit(brute_force, number=10))
    print(brute_force())

    66.96907186399949
    31875000

Let's do some optimization by cancelling the loops when we exceed the boundaries. Actually, I have also realized that choosing 251 and 501 is a little bit arbitrary. For example if a, b, c where something like $332, 333, 335$ that could be a solution and the first range was too low. Then again, we would have realized that as soon as we get back None, so it is okay.

In [2]:

    def brute_force():
        for a in range(1, 251):
            for b in range(a + 1, 501):
                if a + b > 600:
                    break
                for c in range(b + 1, 1001):
                    if a + b + c == 1000 and a * a + b * b == c * c:
                        return a * b * c

    print(timeit.timeit(brute_force, number=10))
    print(brute_force())

    64.4656584559998
    31875000

Big time save. Kappa. Okay, I am stupid. If I have a and b I can calculate c and check if it is a solution.

In [3]:

    def smart_brute_force():
        for a in range(1, 251):
            for b in range(a + 1, 501):
                c = 1000 - a - b
                if a * a + b * b == c * c:
                    return a * b * c

    print(timeit.timeit(smart_brute_force, number=10))
    print(smart_brute_force())

    0.22822808900036762
    31875000

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