Problem

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.

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Problem Description

In a Pythagorean triplet of natural numbers, a < b < c for which,

$$a^2 + b^2 = c^2$$

Example,

$$3^2 + 4^2 = 9 + 16 = 25 = 5^2$$

In this case a + b + c is 12 and a * b * c is 60

Same way, there exists one triplet, where a + b + c = 1000 and we need to find a * b * c of that triplet.

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Approach

Let's approach this problem as finding some triplets where a + b + c = n.

There are as many different mathematical approaches to solving this(I have known only 2. But Wikipedia has more than that 😎).

The straightforward approach is to loop over a and b to check,

$$a^2 + b^2 = (n − a − b)^2$$

And since we have a < b < c, we use the below formula while looping through a and b - a <= (n - 3)/3 & b < (n - a)/2

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Solution

const pythagoreanTriplet = n => {
    let maxProduct = -1;

    for(let a = 1; a <= (n - 3)/3; a++) {
        for (let b = a; b <= (n - a)/2 ; b++) {
            if( (a*a) + (b*b) === (n-a-b)*(n-a-b)) {
                maxProduct = a*b*(n-a-b);
            }
        }
    }
    return maxProduct;
}
console.log(pythagoreanTriplet(1000));

You can find my solution on GitHub 09. My GitHub solution is different from the above. To achieve 100% success(all test cases passed) in Hacker Rank, I tweaked a bit using the formula.

If you have another or a better solution, please leave it in the comments below.

For the other Project Euler Solutions, please follow the series Project Euler Solutions in JS.

Thank you!