feat(problem): 9 - special pythagorean triplet

readme.md

@@ -9,7 +9,7 @@ All of the solutions here are written in rust. [main.rs](src/main.rs) is the hom
 
 ## Challenge Completion
 
-<!-- completed -->8<!-- completed --> out of 100 public challenges completed.
+<!-- completed -->9<!-- completed --> out of 100 public challenges completed.
 
 - [x] 1 - [Multiples of 3 or 5](src/bin/1.rs)
 - [x] 2 - [Even Fibonacci numbers](src/bin/2.rs)
@@ -19,7 +19,7 @@ All of the solutions here are written in rust. [main.rs](src/main.rs) is the hom
 - [x] 6 - [Sum square difference](src/bin/6.rs)
 - [x] 7 - [10001st prime](src/bin/7.rs)
 - [x] 8 - [Largest product in a series](src/bin/8.rs)
-- [ ] 9 - Special Pythagorean triplet
+- [x] 9 - [Special Pythagorean triplet](src/bin/9.rs)
 - [ ] 10 - Summation of primes
 - [ ] 11 - Largest product in a grid
 - [ ] 12 - Highly divisible triangular number
@@ -110,4 +110,4 @@ All of the solutions here are written in rust. [main.rs](src/main.rs) is the hom
 - [ ] 97 - Large non-Mersenne prime
 - [ ] 98 - Anagramic squares
 - [ ] 99 - Largest exponential
-- [ ] 100 - Arranged probability
+- [ ] 100 - Arranged probability

src/bin/9.rs

@@ -0,0 +1,41 @@
+/*
+Problem 9 - Special Pythagorean triplet
+
+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 .
+*/
+
+//! Notes:
+// a + b + c = n
+// c = n - a - b
+
+// a^2 + b^2 = c^2
+// a^2 + b^2 = (n - a - b)^2
+// 0 = 2ab - 2an - 2bn + n^2
+// 2an - n^2 = b(2a - 2n)
+// b = (2an - n^2)/(2a - 2n)
+
+// For a given value a, with a sum of n
+// b = (2an - n^2)/(2a - 2n)
+// c = n - a - b
+
+fn triplet_with_sum(sum: isize) -> (isize, isize, isize) {
+    for a in 1..(sum + 1) {
+        let b: isize = ((2 * a * sum) - sum.pow(2)) / (2 * (a - sum));
+        let c: isize  = sum - a - b;
+
+        if a.pow(2) + b.pow(2) == c.pow(2) {
+            return (a, b, c);
+        }
+    }
+
+    panic!("A triplet could not be found with the sum {sum}!");
+}
+
+fn main() {
+    let (a, b, c) = triplet_with_sum(1000);
+    
+    println!("a = {a}, b = {b}, c = {c} // abc = {}", a * b * c);
+}