Update day 17 solution

day-17/day-17.py

@@ -3,22 +3,22 @@ from pathlib import Path
 
 
 def part_1(input):
+    """The max height is reached by using the maximum start y velocity that doesn't overshoot the
+    landing area. The x velocity doesn't mater here, because x position won't change after x
+    velocity reaches 0. So there will always be an vx that drops straight between the x boundaries.
+    Because the y velocity decreases linear we will reach our starting height after the velocity
+    changed exactly to the negative value of our starting velocity. At this point we have to make
+    sure not to overhoot the landing area, so the next velocity needs to be the same value as the
+    lowest coordinate of the landing area to be able to start with the highest possible starting
+    velocity. This leaves us with a starting velocity of abs(y_min + 1) to reach the maximum
+    possible height. For positive landing coordinates y_max has to be used.
+    """
     result = 0
     _, y = input[0].rstrip().replace('target area: ', '').replace(
         'x=', '').replace('y=', '').split(', ')
-    y_min, y_max = [int(c) for c in y.split('..')]
-    max_valid_height = 0
-    for vy_0 in range(-y_min):
-        vy = vy_0
-        y_pos = 0
-        max_height = 0
-        while y_pos > y_min:
-            y_pos += vy
-            max_height = max(max_height, y_pos)
-            vy -= 1
-            if y_min <= y_pos <= y_max:
-                max_valid_height = max(max_valid_height, max_height)
-    result = max_valid_height
+    y_min, _ = [int(c) for c in y.split('..')]
+    vy0 = abs(y_min + 1)
+    result = vy0 * (vy0 + 1) // 2
     print("Part 1 result:", result)