## \-\-\- Day 24: Never Tell Me The Odds ---

It seems like something is going wrong with the snow-making process. Instead of forming snow, the water that's been absorbed into the air seems to be forming [hail](https://en.wikipedia.org/wiki/Hail)!

Maybe there's something you can do to break up the hailstones?

Due to strong, probably-magical winds, the hailstones are all flying through the air in perfectly linear trajectories. You make a note of each hailstone's _position_ and _velocity_ (your puzzle input). For example:

```
19, 13, 30 @ -2,  1, -2
18, 19, 22 @ -1, -1, -2
20, 25, 34 @ -2, -2, -4
12, 31, 28 @ -1, -2, -1
20, 19, 15 @  1, -5, -3

```

Each line of text corresponds to the position and velocity of a single hailstone. The positions indicate where the hailstones are _right now_ (at time `0`). The velocities are constant and indicate exactly how far each hailstone will move in _one nanosecond_.

Each line of text uses the format `px py pz @ vx vy vz`. For instance, the hailstone specified by `20, 19, 15 @  1, -5, -3` has initial X position `20`, Y position `19`, Z position `15`, X velocity `1`, Y velocity `-5`, and Z velocity `-3`. After one nanosecond, the hailstone would be at `21, 14, 12`.

Perhaps you won't have to do anything. How likely are the hailstones to collide with each other and smash into tiny ice crystals?

To estimate this, consider only the X and Y axes; _ignore the Z axis_. Looking _forward in time_, how many of the hailstones' _paths_ will intersect within a test area? (The hailstones themselves don't have to collide, just test for intersections between the paths they will trace.)

In this example, look for intersections that happen with an X and Y position each at least `7` and at most `27`; in your actual data, you'll need to check a much larger test area. Comparing all pairs of hailstones' future paths produces the following results:

```
Hailstone A: 19, 13, 30 @ -2, 1, -2
Hailstone B: 18, 19, 22 @ -1, -1, -2
Hailstones' paths will cross inside the test area (at x=14.333, y=15.333).

Hailstone A: 19, 13, 30 @ -2, 1, -2
Hailstone B: 20, 25, 34 @ -2, -2, -4
Hailstones' paths will cross inside the test area (at x=11.667, y=16.667).

Hailstone A: 19, 13, 30 @ -2, 1, -2
Hailstone B: 12, 31, 28 @ -1, -2, -1
Hailstones' paths will cross outside the test area (at x=6.2, y=19.4).

Hailstone A: 19, 13, 30 @ -2, 1, -2
Hailstone B: 20, 19, 15 @ 1, -5, -3
Hailstones' paths crossed in the past for hailstone A.

Hailstone A: 18, 19, 22 @ -1, -1, -2
Hailstone B: 20, 25, 34 @ -2, -2, -4
Hailstones' paths are parallel; they never intersect.

Hailstone A: 18, 19, 22 @ -1, -1, -2
Hailstone B: 12, 31, 28 @ -1, -2, -1
Hailstones' paths will cross outside the test area (at x=-6, y=-5).

Hailstone A: 18, 19, 22 @ -1, -1, -2
Hailstone B: 20, 19, 15 @ 1, -5, -3
Hailstones' paths crossed in the past for both hailstones.

Hailstone A: 20, 25, 34 @ -2, -2, -4
Hailstone B: 12, 31, 28 @ -1, -2, -1
Hailstones' paths will cross outside the test area (at x=-2, y=3).

Hailstone A: 20, 25, 34 @ -2, -2, -4
Hailstone B: 20, 19, 15 @ 1, -5, -3
Hailstones' paths crossed in the past for hailstone B.

Hailstone A: 12, 31, 28 @ -1, -2, -1
Hailstone B: 20, 19, 15 @ 1, -5, -3
Hailstones' paths crossed in the past for both hailstones.

```

So, in this example, `2` hailstones' future paths cross inside the boundaries of the test area.

However, you'll need to search a much larger test area if you want to see if any hailstones might collide. Look for intersections that happen with an X and Y position each at least `200000000000000` and at most `400000000000000`. Disregard the Z axis entirely.

Considering only the X and Y axes, check all pairs of hailstones' future paths for intersections. _How many of these intersections occur within the test area?_