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--- Day 12: The N-Body Problem ---
The space near Jupiter is not a very safe place; you need to be careful of a big distracting red
spot, extreme radiation, and a whole lot of moons swirling around. You decide to start by tracking
the four largest moons: [1m[97mIo[0m, [1m[97mEuropa[0m, [1m[97mGanymede[0m, and [1m[97mCallisto[0m.
After a brief scan, you calculate the [1m[97mposition of each moon[0m (your puzzle input). You just need to
[1m[97msimulate their motion[0m so you can avoid them.
Each moon has a 3-dimensional position (x, y, and z) and a 3-dimensional velocity. The position of
each moon is given in your scan; the x, y, and z velocity of each moon starts at 0.
Simulate the motion of the moons in [1m[97mtime steps[0m. Within each time step, first update the velocity of
every moon by applying [1m[97mgravity[0m. Then, once all moons' velocities have been updated, update the
position of every moon by applying [1m[97mvelocity[0m. Time progresses by one step once all of the positions
are updated.
To apply [1m[97mgravity[0m, consider every [1m[97mpair[0m of moons. On each axis (x, y, and z), the velocity of each
moon changes by [1m[97mexactly +1 or -1[0m to pull the moons together. For example, if Ganymede has an x
position of 3, and Callisto has a x position of 5, then Ganymede's x velocity [1m[97mchanges by +1[0m (because
5 > 3) and Callisto's x velocity [1m[97mchanges by -1[0m (because 3 < 5). However, if the positions on a given
axis are the same, the velocity on that axis [1m[97mdoes not change[0m for that pair of moons.
Once all gravity has been applied, apply [1m[97mvelocity[0m: simply add the velocity of each moon to its own
position. For example, if Europa has a position of x=1, y=2, z=3 and a velocity of x=-2, y=0,z=3,
then its new position would be x=-1, y=2, z=6. This process does not modify the velocity of any
moon.
For example, suppose your scan reveals the following positions:
<x=-1, y=0, z=2>
<x=2, y=-10, z=-7>
<x=4, y=-8, z=8>
<x=3, y=5, z=-1>
Simulating the motion of these moons would produce the following:
After 0 steps:
pos=<x=-1, y= 0, z= 2>, vel=<x= 0, y= 0, z= 0>
pos=<x= 2, y=-10, z=-7>, vel=<x= 0, y= 0, z= 0>
pos=<x= 4, y= -8, z= 8>, vel=<x= 0, y= 0, z= 0>
pos=<x= 3, y= 5, z=-1>, vel=<x= 0, y= 0, z= 0>
After 1 step:
pos=<x= 2, y=-1, z= 1>, vel=<x= 3, y=-1, z=-1>
pos=<x= 3, y=-7, z=-4>, vel=<x= 1, y= 3, z= 3>
pos=<x= 1, y=-7, z= 5>, vel=<x=-3, y= 1, z=-3>
pos=<x= 2, y= 2, z= 0>, vel=<x=-1, y=-3, z= 1>
After 2 steps:
pos=<x= 5, y=-3, z=-1>, vel=<x= 3, y=-2, z=-2>
pos=<x= 1, y=-2, z= 2>, vel=<x=-2, y= 5, z= 6>
pos=<x= 1, y=-4, z=-1>, vel=<x= 0, y= 3, z=-6>
pos=<x= 1, y=-4, z= 2>, vel=<x=-1, y=-6, z= 2>
After 3 steps:
pos=<x= 5, y=-6, z=-1>, vel=<x= 0, y=-3, z= 0>
pos=<x= 0, y= 0, z= 6>, vel=<x=-1, y= 2, z= 4>
pos=<x= 2, y= 1, z=-5>, vel=<x= 1, y= 5, z=-4>
pos=<x= 1, y=-8, z= 2>, vel=<x= 0, y=-4, z= 0>
After 4 steps:
pos=<x= 2, y=-8, z= 0>, vel=<x=-3, y=-2, z= 1>
pos=<x= 2, y= 1, z= 7>, vel=<x= 2, y= 1, z= 1>
pos=<x= 2, y= 3, z=-6>, vel=<x= 0, y= 2, z=-1>
pos=<x= 2, y=-9, z= 1>, vel=<x= 1, y=-1, z=-1>
After 5 steps:
pos=<x=-1, y=-9, z= 2>, vel=<x=-3, y=-1, z= 2>
pos=<x= 4, y= 1, z= 5>, vel=<x= 2, y= 0, z=-2>
pos=<x= 2, y= 2, z=-4>, vel=<x= 0, y=-1, z= 2>
pos=<x= 3, y=-7, z=-1>, vel=<x= 1, y= 2, z=-2>
After 6 steps:
pos=<x=-1, y=-7, z= 3>, vel=<x= 0, y= 2, z= 1>
pos=<x= 3, y= 0, z= 0>, vel=<x=-1, y=-1, z=-5>
pos=<x= 3, y=-2, z= 1>, vel=<x= 1, y=-4, z= 5>
pos=<x= 3, y=-4, z=-2>, vel=<x= 0, y= 3, z=-1>
After 7 steps:
pos=<x= 2, y=-2, z= 1>, vel=<x= 3, y= 5, z=-2>
pos=<x= 1, y=-4, z=-4>, vel=<x=-2, y=-4, z=-4>
pos=<x= 3, y=-7, z= 5>, vel=<x= 0, y=-5, z= 4>
pos=<x= 2, y= 0, z= 0>, vel=<x=-1, y= 4, z= 2>
After 8 steps:
pos=<x= 5, y= 2, z=-2>, vel=<x= 3, y= 4, z=-3>
pos=<x= 2, y=-7, z=-5>, vel=<x= 1, y=-3, z=-1>
pos=<x= 0, y=-9, z= 6>, vel=<x=-3, y=-2, z= 1>
pos=<x= 1, y= 1, z= 3>, vel=<x=-1, y= 1, z= 3>
After 9 steps:
pos=<x= 5, y= 3, z=-4>, vel=<x= 0, y= 1, z=-2>
pos=<x= 2, y=-9, z=-3>, vel=<x= 0, y=-2, z= 2>
pos=<x= 0, y=-8, z= 4>, vel=<x= 0, y= 1, z=-2>
pos=<x= 1, y= 1, z= 5>, vel=<x= 0, y= 0, z= 2>
After 10 steps:
pos=<x= 2, y= 1, z=-3>, vel=<x=-3, y=-2, z= 1>
pos=<x= 1, y=-8, z= 0>, vel=<x=-1, y= 1, z= 3>
pos=<x= 3, y=-6, z= 1>, vel=<x= 3, y= 2, z=-3>
pos=<x= 2, y= 0, z= 4>, vel=<x= 1, y=-1, z=-1>
Then, it might help to calculate the [1m[97mtotal energy in the system[0m. The total energy for a single moon
is its [1m[97mpotential energy[0m multiplied by its [1m[97mkinetic energy[0m. A moon's [1m[97mpotential energy[0m is the sum of
the absolute values of its x, y, and z position coordinates. A moon's [1m[97mkinetic energy[0m is the sum of
the absolute values of its velocity coordinates. Below, each line shows the calculations for a
moon's potential energy (pot), kinetic energy (kin), and total energy:
Energy after 10 steps:
pot: 2 + 1 + 3 = 6; kin: 3 + 2 + 1 = 6; total: 6 * 6 = 36
pot: 1 + 8 + 0 = 9; kin: 1 + 1 + 3 = 5; total: 9 * 5 = 45
pot: 3 + 6 + 1 = 10; kin: 3 + 2 + 3 = 8; total: 10 * 8 = 80
pot: 2 + 0 + 4 = 6; kin: 1 + 1 + 1 = 3; total: 6 * 3 = 18
Sum of total energy: 36 + 45 + 80 + 18 = [1m[97m179[0m
In the above example, adding together the total energy for all moons after 10 steps produces the
total energy in the system, [1m[97m179[0m.
Here's a second example:
<x=-8, y=-10, z=0>
<x=5, y=5, z=10>
<x=2, y=-7, z=3>
<x=9, y=-8, z=-3>
Every ten steps of simulation for 100 steps produces:
After 0 steps:
pos=<x= -8, y=-10, z= 0>, vel=<x= 0, y= 0, z= 0>
pos=<x= 5, y= 5, z= 10>, vel=<x= 0, y= 0, z= 0>
pos=<x= 2, y= -7, z= 3>, vel=<x= 0, y= 0, z= 0>
pos=<x= 9, y= -8, z= -3>, vel=<x= 0, y= 0, z= 0>
After 10 steps:
pos=<x= -9, y=-10, z= 1>, vel=<x= -2, y= -2, z= -1>
pos=<x= 4, y= 10, z= 9>, vel=<x= -3, y= 7, z= -2>
pos=<x= 8, y=-10, z= -3>, vel=<x= 5, y= -1, z= -2>
pos=<x= 5, y=-10, z= 3>, vel=<x= 0, y= -4, z= 5>
After 20 steps:
pos=<x=-10, y= 3, z= -4>, vel=<x= -5, y= 2, z= 0>
pos=<x= 5, y=-25, z= 6>, vel=<x= 1, y= 1, z= -4>
pos=<x= 13, y= 1, z= 1>, vel=<x= 5, y= -2, z= 2>
pos=<x= 0, y= 1, z= 7>, vel=<x= -1, y= -1, z= 2>
After 30 steps:
pos=<x= 15, y= -6, z= -9>, vel=<x= -5, y= 4, z= 0>
pos=<x= -4, y=-11, z= 3>, vel=<x= -3, y=-10, z= 0>
pos=<x= 0, y= -1, z= 11>, vel=<x= 7, y= 4, z= 3>
pos=<x= -3, y= -2, z= 5>, vel=<x= 1, y= 2, z= -3>
After 40 steps:
pos=<x= 14, y=-12, z= -4>, vel=<x= 11, y= 3, z= 0>
pos=<x= -1, y= 18, z= 8>, vel=<x= -5, y= 2, z= 3>
pos=<x= -5, y=-14, z= 8>, vel=<x= 1, y= -2, z= 0>
pos=<x= 0, y=-12, z= -2>, vel=<x= -7, y= -3, z= -3>
After 50 steps:
pos=<x=-23, y= 4, z= 1>, vel=<x= -7, y= -1, z= 2>
pos=<x= 20, y=-31, z= 13>, vel=<x= 5, y= 3, z= 4>
pos=<x= -4, y= 6, z= 1>, vel=<x= -1, y= 1, z= -3>
pos=<x= 15, y= 1, z= -5>, vel=<x= 3, y= -3, z= -3>
After 60 steps:
pos=<x= 36, y=-10, z= 6>, vel=<x= 5, y= 0, z= 3>
pos=<x=-18, y= 10, z= 9>, vel=<x= -3, y= -7, z= 5>
pos=<x= 8, y=-12, z= -3>, vel=<x= -2, y= 1, z= -7>
pos=<x=-18, y= -8, z= -2>, vel=<x= 0, y= 6, z= -1>
After 70 steps:
pos=<x=-33, y= -6, z= 5>, vel=<x= -5, y= -4, z= 7>
pos=<x= 13, y= -9, z= 2>, vel=<x= -2, y= 11, z= 3>
pos=<x= 11, y= -8, z= 2>, vel=<x= 8, y= -6, z= -7>
pos=<x= 17, y= 3, z= 1>, vel=<x= -1, y= -1, z= -3>
After 80 steps:
pos=<x= 30, y= -8, z= 3>, vel=<x= 3, y= 3, z= 0>
pos=<x= -2, y= -4, z= 0>, vel=<x= 4, y=-13, z= 2>
pos=<x=-18, y= -7, z= 15>, vel=<x= -8, y= 2, z= -2>
pos=<x= -2, y= -1, z= -8>, vel=<x= 1, y= 8, z= 0>
After 90 steps:
pos=<x=-25, y= -1, z= 4>, vel=<x= 1, y= -3, z= 4>
pos=<x= 2, y= -9, z= 0>, vel=<x= -3, y= 13, z= -1>
pos=<x= 32, y= -8, z= 14>, vel=<x= 5, y= -4, z= 6>
pos=<x= -1, y= -2, z= -8>, vel=<x= -3, y= -6, z= -9>
After 100 steps:
pos=<x= 8, y=-12, z= -9>, vel=<x= -7, y= 3, z= 0>
pos=<x= 13, y= 16, z= -3>, vel=<x= 3, y=-11, z= -5>
pos=<x=-29, y=-11, z= -1>, vel=<x= -3, y= 7, z= 4>
pos=<x= 16, y=-13, z= 23>, vel=<x= 7, y= 1, z= 1>
Energy after 100 steps:
pot: 8 + 12 + 9 = 29; kin: 7 + 3 + 0 = 10; total: 29 * 10 = 290
pot: 13 + 16 + 3 = 32; kin: 3 + 11 + 5 = 19; total: 32 * 19 = 608
pot: 29 + 11 + 1 = 41; kin: 3 + 7 + 4 = 14; total: 41 * 14 = 574
pot: 16 + 13 + 23 = 52; kin: 7 + 1 + 1 = 9; total: 52 * 9 = 468
Sum of total energy: 290 + 608 + 574 + 468 = [1m[97m1940[0m
[1m[97mWhat is the total energy in the system[0m after simulating the moons given in your scan for 1000 steps?
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