You finally decode the Elves' message. `HI`, the message says. You continue searching for the sleigh keys. Ahead of you is what appears to be a large [ocean trench](https://en.wikipedia.org/wiki/Oceanic_trench). Could the keys have fallen into it? You'd better send a probe to investigate. The probe launcher on your submarine can fire the probe with any [integer](https://en.wikipedia.org/wiki/Integer) velocity in the `x` (forward) and `y` (upward, or downward if negative) directions. For example, an initial `x,y` velocity like `0,10` would fire the probe straight up, while an initial velocity like `10,-1` would fire the probe forward at a slight downward angle. The probe's `x,y` position starts at `0,0`. Then, it will follow some trajectory by moving in *steps*. On each step, these changes occur in the following order: * The probe's `x` position increases by its `x` velocity. * The probe's `y` position increases by its `y` velocity. * Due to drag, the probe's `x` velocity changes by `1` toward the value `0`; that is, it decreases by `1` if it is greater than `0`, increases by `1` if it is less than `0`, or does not change if it is already `0`. * Due to gravity, the probe's `y` velocity decreases by `1`. For the probe to successfully make it into the trench, the probe must be on some trajectory that causes it to be within a *target area* after any step. The submarine computer has already calculated this target area (your puzzle input). For example: ``` target area: x=20..30, y=-10..-5 ``` This target area means that you need to find initial `x,y` velocity values such that after any step, the probe's `x` position is at least `20` and at most `30`, *and* the probe's `y` position is at least `-10` and at most `-5`. Given this target area, one initial velocity that causes the probe to be within the target area after any step is `7,2`: ``` .............#....#............ .......#..............#........ ............................... S........................#..... ............................... ............................... ...........................#... ............................... ....................TTTTTTTTTTT ....................TTTTTTTTTTT ....................TTTTTTTT#TT ....................TTTTTTTTTTT ....................TTTTTTTTTTT ....................TTTTTTTTTTT ``` In this diagram, `S` is the probe's initial position, `0,0`. The `x` coordinate increases to the right, and the `y` coordinate increases upward. In the bottom right, positions that are within the target area are shown as `T`. After each step (until the target area is reached), the position of the probe is marked with `#`. (The bottom-right `#` is both a position the probe reaches and a position in the target area.) Another initial velocity that causes the probe to be within the target area after any step is `6,3`: ``` ...............#..#............ ...........#........#.......... ............................... ......#..............#......... ............................... ............................... S....................#......... ............................... ............................... ............................... .....................#......... ....................TTTTTTTTTTT ....................TTTTTTTTTTT ....................TTTTTTTTTTT ....................TTTTTTTTTTT ....................T#TTTTTTTTT ....................TTTTTTTTTTT ``` Another one is `9,0`: ``` S........#..................... .................#............. ............................... ........................#...... ............................... ....................TTTTTTTTTTT ....................TTTTTTTTTT# ....................TTTTTTTTTTT ....................TTTTTTTTTTT ....................TTTTTTTTTTT ....................TTTTTTTTTTT ``` One initial velocity that *doesn't* cause the probe to be within the target area after any step is `17,-4`: ``` S.............................................................. ............................................................... ............................................................... ............................................................... .................#............................................. ....................TTTTTTTTTTT................................ ....................TTTTTTTTTTT................................ ....................TTTTTTTTTTT................................ ....................TTTTTTTTTTT................................ ....................TTTTTTTTTTT..#............................. ....................TTTTTTTTTTT................................ ............................................................... ............................................................... ............................................................... ............................................................... ................................................#.............. ............................................................... ............................................................... ............................................................... ............................................................... ............................................................... ............................................................... ..............................................................# ``` The probe appears to pass through the target area, but is never within it after any step. Instead, it continues down and to the right - only the first few steps are shown. If you're going to fire a highly scientific probe out of a super cool probe launcher, you might as well do it with *style*. How high can you make the probe go while still reaching the target area? In the above example, using an initial velocity of `6,9` is the best you can do, causing the probe to reach a maximum `y` position of `*45*`. (Any higher initial `y` velocity causes the probe to overshoot the target area entirely.) Find the initial velocity that causes the probe to reach the highest `y` position and still eventually be within the target area after any step. *What is the highest `y` position it reaches on this trajectory?* To begin, [get your puzzle input](17/input). Answer: