24.worksheet.sc

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--- 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!

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?

--- Part Two ---
Upon further analysis, it doesn't seem like any hailstones will naturally
collide. It's up to you to fix that!

You find a rock on the ground nearby. While it seems extremely unlikely, if you
throw it just right, you should be able to hit every hailstone in a single
throw!

You can use the probably-magical winds to reach any integer position you like
and to propel the rock at any integer velocity. Now including the Z axis in your
calculations, if you throw the rock at time 0, where do you need to be so that
the rock perfectly collides with every hailstone? Due to probably-magical
inertia, the rock won't slow down or change direction when it collides with a
hailstone.

In the example above, you can achieve this by moving to position 24, 13, 10 and
throwing the rock at velocity -3, 1, 2. If you do this, you will hit every
hailstone as follows:

Hailstone: 19, 13, 30 @ -2, 1, -2
Collision time: 5
Collision position: 9, 18, 20

Hailstone: 18, 19, 22 @ -1, -1, -2
Collision time: 3
Collision position: 15, 16, 16

Hailstone: 20, 25, 34 @ -2, -2, -4
Collision time: 4
Collision position: 12, 17, 18

Hailstone: 12, 31, 28 @ -1, -2, -1
Collision time: 6
Collision position: 6, 19, 22

Hailstone: 20, 19, 15 @ 1, -5, -3
Collision time: 1
Collision position: 21, 14, 12

Above, each hailstone is identified by its initial position and its velocity.
Then, the time and position of that hailstone's collision with your rock are
given.

After 1 nanosecond, the rock has exactly the same position as one of the
hailstones, obliterating it into ice dust! Another hailstone is smashed to bits
two nanoseconds after that. After a total of 6 nanoseconds, all of the
hailstones have been destroyed.

So, at time 0, the rock needs to be at X position 24, Y position 13, and Z
position 10. Adding these three coordinates together produces 47. (Don't add any
coordinates from the rock's velocity.)

Determine the exact position and velocity the rock needs to have at time 0 so
that it perfectly collides with every hailstone. What do you get if you add up
the X, Y, and Z coordinates of that initial position?
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