Particle Swarm

$---$ Part One $---$

You are trying to simulate a particle swarm for an AI project.

You have access to a text file with each particle in order $($starting with particle $0,$ then particle $1,$

particle $2,$ and so on$).$ For each particle, it provides the X$,$ Y$,$ and Z coordinates for the particle's

position $($p$),$ velocity $($v$),$ and acceleration $($a$),$ each in the format $.$

Each tick, all particles are updated simultaneously. A particle's properties are updated in the following

order:

$$ Increase the X velocity by the X acceleration.

$$ Increase the Y velocity by the Y acceleration.

$$ Increase the Z velocity by the Z acceleration.

$$ Increase the X position by the X velocity.

$$ Increase the Y position by the Y velocity.

$$ Increase the Z position by the Z velocity.

You would like to know which particle will stay closest to position $<0,0,0>$ in the long term.

Measure this using the Manhattan distance, which in this situation is simply the sum of the absolute

values of a particle's X$,$ Y$,$ and Z position.

For example, suppose you are only given two particles, both of which stay entirely on the X$-$axis $($for

simplicity$).$ Drawing the current states of particles $0$ and $1($in that order$)$ with an adjacent a number

line and diagram of current X positions $($marked in parentheses$),$ the following would take place:

p$=<3,0,0>,$ v$=<2,0,0>,$ a$=<-1,0,0>-4-3-2-101234$

p$=<4,0,0>,$ v$=<0,0,0>,$ a$=<-2,0,0>(0)(1)$

p$=<4,0,0>,$ v$=<1,0,0>,$ a$=<-1,0,0>-4-3-2-101234$

p$=<2,0,0>,$ v$=<-2,0,0>,$ a$=<-2,0,0>(1)(0)$

p$=<4,0,0>,$ v$=<0,0,0>,$ a$=<-1,0,0>-4-3-2-101234$

p$=<-2,0,0>,$ v$=<-4,0,0>,$ a$=<-2,0,0>(1)(0)$

p$=<3,0,0>,$ v$=<-1,0,0>,$ a$=<-1,0,0>-4-3-2-101234$

p$=<-8,0,0>,$ v$=<-6,0,0>,$ a$=<-2,0,0>(0)$

At this point, particle $1$ will never be closer to $<0,0,0>$ than particle $0,$ and so$,$ in the long run,

particle $0$ will stay closest.

Which particle will stay closest to position $<0,0,0>$ in the long term?

$---$ Part Two $---$

To simplify the problem further, you would like to remove any particles that collide. Particles collide if

their positions ever exactly match. Because particles are updated simultaneously, more than two

particles can collide at the same time and place. Once particles collide, they are removed and cannot

collide with anything else after that tick.

For example:

p$=<-6,0,0>,$ v$=<3,0,0>,$ a$=<0,0,0>$

p$=<-4,0,0>,$ v$=<2,0,0>,$ a$=<0,0,0>-6-5-4-3-2-10123$

p$=<-2,0,0>,$ v$=<1,0,0>,$ a$=<0,0,0>(0)(1)(2)(3)$

p$=<3,0,0>,$ v$=<-1,0,0>,$ a$=<0,0,0>$

p$=<-3,0,0>,$ v$=<3,0,0>,$ a$=<0,0,0>$

p$=<-2,0,0>,$ v$=<2,0,0>,$ a$=<0,0,0>-6-5-4-3-2-10123$

p$=<-1,0,0>,$ v$=<1,0,0>,$ a$=<0,0,0>(0)(1)(2)(3)$

p$=<2,0,0>,$ v$=<-1,0,0>,$ a$=<0,0,0>$

p$=<0,0,0>,$ v$=<3,0,0>,$ a$=<0,0,0>$

p$=<0,0,0>,$ v$=<2,0,0>,$ a$=<0,0,0>-6-5-4-3-2-10123$

p$=<0,0,0>,$ v$=<1,0,0>,$ a$=<0,0,0>$ X $(3)$

p$=<1,0,0>,$ v$=<-1,0,0>,$ a$=<0,0,0>$

$------$destroyed by collision$------$

$------$destroyed by collision$-------6-5-4-3-2-10123$

$------$destroyed by collision$------(3)$

p$=<0,0,0>,$ v$=<-1,0,0>,$ a$=<0,0,0>$

In this example, particles $0,1,$ and $2$ are simultaneously destroyed at the time and place marked X$.$ On

the next tick, particle $3$ passes through unharmed.

How many particles are left after all collisions are resolved?