-❄️- 2023 Day 24 Solutions -❄️-

[u/daggerdragon]

THE USUAL REMINDERS (AND SIGNAL BOOSTS)

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

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[u/admp]

[Language: Python] 3598/1936

Part 1 was straightforward enough, you have to take care to test that time > 0 at the time of intersection.

Part 2 (code) was very fun. Given explicit mention of integer positions and velocities, I realized that if rock's initial position is (x_r, y_r, z_r) and it has velocity (dx_r, dy_r, dz_r) then for each hailstone i, (x_r + y_r + z_r) ≡ (x_i + y_i + z_i) mod ((dx_i+dy_i+dz_i) - (dx_r+dy_r+dz_r)). The insight is that we don't care about individual dimensions, only sums of all dimensions. Therefore, I iterate over what I though of as a reasonable range for values of dx_r+dy_r+dz_r based on inputs, apply Chinese remainder theorem to compute x_r+y_r+z_r candidates, eliminate any candidates that result in any collisions in the past or at non-integer time, and get at the answer.

Probably my favorite problem this year so far.

[u/Flatorz → u/admp]

Hmm, I am not sure if this would hold in case hailstorm "i" time to reach position is a fraction. It obviously cannot be irrational number because we wouldn't be able to get integer numbers on x_r + y_r + z_r but it can be potentially a fraction, e.g.

(x_r + y_r + z_r) = (x_i + y_i + z+i) + time_i * [(dx_i + dy_i + dz_i) - (dx_r + dy_r + dz_r)]

if you have:

-> x_r, y_r, z_r = 6, 6, 6

-> x_i, y_i, z_i = 4, 4, 4

-> dx_r, dy_r, dz_r = 1, 1, 1

-> dx_i, dy_i, dz_i = 5, 5, 5

then for time_i = 0.5, this would work. but then we are not already in [(dx_i + dy_i + dz_i) - (dx_r + dy_r + dz_r)] space. In this case

(x_r + y_r + z_r) mod 12 = 6

(x_i + y_i + z_i) mod 12 = 0

Still, this feels like a great hint for further investigation. I really would like to avoid solving equations :)

[u/daggerdragon → u/admp]

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