This one was wild. At first I started visualling it but wasn't seeing anything. I went up to almost 80 iterations before giving up. I then went with the "no overlapping points thing" but didn't think that was working for me either.

It was iteration 7,753. f's sake man.

I used the a modulo operation and an if check to skip ahead to the final state. It was also tricky b/c the grid is larger than I could get my console to work with.

F#

(just the important bits)

type Velocity = Point2I
[<Struct>]type Robot = {Position:Point2I; Velocity:Velocity}
type Boundary = {MaxX:int; MaxY: int}
let move ticks boundary robot =
    let newX = ((robot.Position.X + (robot.Velocity.X * ticks)) % boundary.MaxX) |> fun x -> if x < 0 then boundary.MaxX + x else x
    let newY = ((robot.Position.Y + (robot.Velocity.Y * ticks)) % boundary.MaxY) |> fun x -> if x < 0 then boundary.MaxY + x else x
    { robot with Position = {X = newX; Y = newY} }
    
let toQuadrantScore boundary robots =
    let removeX = ((boundary.MaxX |> float) / 2.0) |> int
    let removeY = ((boundary.MaxY |> float) / 2.0) |> int
    ((0,0,0,0), robots)
    ||> Seq.fold(fun (a,b,c,d) robot ->
        if robot.Position.X < removeX && robot.Position.Y < removeY then (a+1,b,c,d)
        else if robot.Position.X > removeX && robot.Position.Y < removeY then (a,b+1,c,d)
        else if robot.Position.X < removeX && robot.Position.Y > removeY then (a,b,c+1,d)
        else if (robot.Position.X > removeX && robot.Position.Y > removeY) then (a,b,c,d+1)
        else (a,b,c,d)
        )
    |> fun (a,b,c,d) -> a*b*c*d


let part1impl boundary ticks robots =
    robots
    |> Seq.map(move ticks boundary)
    |> toQuadrantScore boundary
    
let part1 input =
    (read input parse)
    |> part1impl {MaxX = 101; MaxY = 103 } 100


let part2 input =
    let robots = (read input parse) |> Array.ofSeq
    let boundary = {MaxX = 101; MaxY = 103 }
    
    // I'll steal the no overlapping robots approach
    // since I'll just be iterating through, I'll do batches of 100 numbers each in parallel and then draw it
    // up to 100 batches
    [0..100]
    |> List.find (fun batch ->
        try
            seq {0..100}
            |> Seq.pick (fun t ->
                // I could use PSeq here, but I'd have to remove the console stuff as the locking and fighting for the console in parallel really slows it
                let ticks = batch * 100 + t
                Console.Clear()
                Console.SetCursorPosition(0,0)
                Console.Write(ticks)
                
                let count =
                    robots
                    |> PSeq.map (move ticks boundary)
                    |> PSeq.distinctBy _.Position
                    |> PSeq.length
                    
                
                    
                if count = 500 then
                    ... write to file, console