# Project Euler: #9 - Special Pythagorean triplet

### **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`.

---

### **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.

---

### **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](https://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples) 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`

---

### **Solution**

```javascript
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](https://github.com/sansk/project-euler-solutions-javascript/tree/main/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.**](https://theintrovertcoder.hashnode.dev/series/project-euler)

Thank you!