2024 - Day 13 - added some comments on the equations

src/main/kotlin/no/rodland/advent_2024/Day13.kt

@@ -19,8 +19,22 @@ class Day13(val input: List<String>) : Day<Long, Long, List<Day13.Machine>> {
     }
 
     data class Machine(val a: P, val b: P, val p: P) {
-        val cb = (p.y * a.x - p.x * a.y) / (a.x * b.y - a.y * b.x)
-        val ca = (p.x - cb * b.x) / a.x
+        // Button A: X+94, Y+34
+        // Button B: X+22, Y+67
+        // Prize: X=8400, Y=5400
+
+        // 1: a.x*ca + b.x*cb = p.x
+        // 2: a.y*ca + b.y*cb = p.y
+
+        // 1: ca = (p.x - b.x*cb) / a.x
+        // 2+1: a.y*((p.x - b.x*cb) / a.x) + b.y*cb = p.y
+        //      b.y*cb  - a.y/a.x * b.x*cb  = p.y - a.y/a.x * p.x
+        //      cb * (b.y - a.y/a.x * b.x) = p.y - a.y/a.x * p.x
+        //      cb = (p.y - a.y/a.x * p.x) /  (b.y - a.y/a.x * b.x)
+        //      cb = (a.x * p.y - a.y * p.x) / (a.x * b.y - a.y * b.x)
+
+        val cb = (a.x * p.y - a.y * p.x) / (a.x * b.y - a.y * b.x)
+        val ca = (p.x - b.x * cb) / a.x
         val ok = (cb * b.x + ca * a.x == p.x) && (cb * b.y + ca * a.y == p.y)
     }