148 lines
7.5 KiB
Markdown
148 lines
7.5 KiB
Markdown
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# 2021 Day 17: Trick Shot
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Copyright (c) Eric Wastl
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#### [Direct Link](https://adventofcode.com/2021/day/17)
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## Part 1
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You finally decode the Elves' message. HI, the message says. You continue searching for the sleigh keys.
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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.
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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.
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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:
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- The probe's `x` position increases by its `x` velocity.
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- The probe's `y` position increases by its `y` velocity.
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- 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`.
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- Due to gravity, the probe's `y` velocity decreases by `1`.
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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:
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```
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target area: x=20..30, y=-10..-5
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```
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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`.
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Given this target area, one initial velocity that causes the probe to be within the target area after any step is `7,2`:
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```
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.............#....#............
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.......#..............#........
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...............................
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S........................#.....
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...............................
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...............................
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...........................#...
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...............................
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....................TTTTTTTTTTT
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....................TTTTTTTTTTT
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....................TTTTTTTT#TT
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....................TTTTTTTTTTT
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....................TTTTTTTTTTT
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....................TTTTTTTTTTT
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```
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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.)
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Another initial velocity that causes the probe to be within the target area after any step is `6,3`:
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```
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...............#..#............
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...........#........#..........
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...............................
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......#..............#.........
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...............................
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...............................
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S....................#.........
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...............................
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...............................
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...............................
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.....................#.........
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....................TTTTTTTTTTT
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....................TTTTTTTTTTT
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....................TTTTTTTTTTT
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....................TTTTTTTTTTT
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....................T#TTTTTTTTT
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....................TTTTTTTTTTT
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```
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Another one is `9,0`:
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```
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S........#.....................
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.................#.............
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...............................
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........................#......
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...............................
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....................TTTTTTTTTTT
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....................TTTTTTTTTT#
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....................TTTTTTTTTTT
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....................TTTTTTTTTTT
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....................TTTTTTTTTTT
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....................TTTTTTTTTTT
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```
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One initial velocity that **doesn't** cause the probe to be within the target area after any step is `17,-4`:
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```
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S..............................................................
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...............................................................
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...............................................................
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...............................................................
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.................#.............................................
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....................TTTTTTTTTTT................................
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....................TTTTTTTTTTT................................
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....................TTTTTTTTTTT................................
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....................TTTTTTTTTTT................................
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....................TTTTTTTTTTT..#.............................
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....................TTTTTTTTTTT................................
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...............................................................
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...............................................................
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...............................................................
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...............................................................
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................................................#..............
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...............................................................
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...............................................................
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...............................................................
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...............................................................
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...............................................................
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...............................................................
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..............................................................#
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```
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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.
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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?
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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.)
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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?**
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## Part 2
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Maybe a fancy trick shot isn't the best idea; after all, you only have one probe, so you had better not miss.
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To get the best idea of what your options are for launching the probe, you need to find **every initial velocity** that causes the probe to eventually be within the target area after any step.
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In the above example, there are **`112`** different initial velocity values that meet these criteria:
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```
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23,-10 25,-9 27,-5 29,-6 22,-6 21,-7 9,0 27,-7 24,-5
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25,-7 26,-6 25,-5 6,8 11,-2 20,-5 29,-10 6,3 28,-7
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8,0 30,-6 29,-8 20,-10 6,7 6,4 6,1 14,-4 21,-6
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26,-10 7,-1 7,7 8,-1 21,-9 6,2 20,-7 30,-10 14,-3
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20,-8 13,-2 7,3 28,-8 29,-9 15,-3 22,-5 26,-8 25,-8
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25,-6 15,-4 9,-2 15,-2 12,-2 28,-9 12,-3 24,-6 23,-7
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25,-10 7,8 11,-3 26,-7 7,1 23,-9 6,0 22,-10 27,-6
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8,1 22,-8 13,-4 7,6 28,-6 11,-4 12,-4 26,-9 7,4
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24,-10 23,-8 30,-8 7,0 9,-1 10,-1 26,-5 22,-9 6,5
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7,5 23,-6 28,-10 10,-2 11,-1 20,-9 14,-2 29,-7 13,-3
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23,-5 24,-8 27,-9 30,-7 28,-5 21,-10 7,9 6,6 21,-5
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27,-10 7,2 30,-9 21,-8 22,-7 24,-9 20,-6 6,9 29,-5
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8,-2 27,-8 30,-5 24,-7
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```
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**How many distinct initial velocity values cause the probe to be within the target area after any step?**
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