It's gradually coming back to me. The Haskell Complex type doesn't work particularly nicely as an integer, plus the definition of division is more like "scale", so I just went with my own type.

Then I forgot which of div and quot I should use, and kept getting nearly the right answer :/

import Data.Ix  

data CNum = CNum !Integer !Integer  

instance Show CNum where  
  show (CNum x y) = "[" ++ show x ++ "," ++ show y ++ "]"  

cadd, cmul, cdiv :: CNum -> CNum -> CNum  
(CNum x1 y1) `cadd` (CNum x2 y2) = CNum (x1 + x2) (y1 + y2)  
(CNum x1 y1) `cmul` (CNum x2 y2) = CNum (x1 * x2 - y1 * y2) (x1 * y2 + y1 * x2)  
(CNum x1 y1) `cdiv` (CNum x2 y2) = CNum (x1 `quot` x2) (y1 `quot` y2)  

part1 a = iterate op (CNum 0 0) !! 3  
  where  
    op x = ((x `cmul` x) `cdiv` CNum 10 10) `cadd` a  

countEngraved = length . filter engrave  
  where  
    engrave p =  
      let rs = take 100 $ tail $ iterate (op p) (CNum 0 0)  
       in all (\(CNum x y) -> abs x <= 1000000 && abs y <= 1000000) rs  
    op p r = ((r `cmul` r) `cdiv` CNum 100000 100000) `cadd` p  

part2 a =  
  countEngraved  
    . map (\(y, x) -> a `cadd` CNum (x * 10) (y * 10))  
    $ range ((0, 0), (100, 100))  

part3 a =  
  countEngraved  
    . map (\(y, x) -> a `cadd` CNum x y)  
    $ range ((0, 0), (1000, 1000))  

main = do  
  print $ part1 $ CNum 164 56  
  print $ part2 $ CNum (-21723) 67997  
  print $ part3 $ CNum (-21723) 67997  

I struggled for a long time because I had nearly the correct results. I had to switch div with quot.

This puzzle was fun. If you have a visualization, it's even cooler. (It's a fractal)

::: spoiler Haskell Code

{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE PatternSynonyms #-}
{-# OPTIONS_GHC -Wall #-}
module Main (main) where
import Text.Read (ReadPrec, Read (readPrec))
import Data.Functor ((<&>))
import Data.Text (pattern (:<), Text)
import qualified Data.Text as Text
import qualified Data.Text.IO as TextIO
import Control.Monad ((<$!>))
import Control.Arrow ((<<<))

newtype Complex = Complex (Int, Int)

instance Read Complex where
  readPrec :: ReadPrec Complex
  readPrec = readPrec <&> \case
    [a, b] -> Complex (a, b)
    _ -> undefined

instance Show Complex where
  show :: Complex -> String
  show (Complex (a, b))= show [a, b]

readAEquals :: Text -> Complex
readAEquals ('A' :< '=':< rest) = read $ Text.unpack rest
readAEquals _ = undefined


-- >>> Complex (1, 1) `add` Complex (2, 2)
-- [3,3]

add :: Complex -> Complex -> Complex
(Complex (x1, y1)) `add` (Complex (x2, y2)) = Complex (x1 + x2, y1 + y2)

-- >>> Complex (2, 5) `times` Complex (5, 7)
-- [-25,-11]

times :: Complex -> Complex -> Complex
(Complex (x1, y1)) `times` (Complex (x2, y2)) = Complex (x1 * x2 - y1 * y2, x1 * y2 + x2 * y1)

dividedBy :: Complex -> Complex -> Complex
(Complex (x1, y1)) `dividedBy` (Complex (x2, y2)) = Complex (x1 `quot` x2, y1 `quot` y2)

step :: Complex -> Complex -> Complex
step a r = let
 r1 = r `times` r
 r2 = r1 `dividedBy` Complex (10, 10)
 r3 = r2 `add` a
 in r3

zero :: Complex
zero = Complex (0, 0)

part1 :: Complex -> Complex
part1 a = iterate (step a) (Complex (0, 0)) !! 3

shouldBeEngraved :: Complex -> Bool
shouldBeEngraved complexPoint = let

  cycleStep :: Complex -> Complex -> Complex
  cycleStep point r = let
    r2 = r `times` r
    r3 = r2 `dividedBy` Complex (100000, 100000)
    in point `add` r3

  inRange x = x <= 1000000 && x >= -1000000


  in all (\ (Complex (x, y)) -> inRange x && inRange y)
    <<< take 101
    <<< iterate (cycleStep complexPoint)
    $ zero

-- >>> shouldBeEngraved $ Complex (35630,-64880)
-- True
-- >>> shouldBeEngraved $ Complex (35460, -64910)
-- False
-- >>> shouldBeEngraved $ Complex (35630, -64830)
-- False

part2 :: Complex -> Int
part2 (Complex (xA, yA)) = let

    xB = xA + 1000
    yB = yA + 1000

  in length . filter shouldBeEngraved $ do
    x <- [xA, xA+10.. xB]
    y <- [yA, yA+10.. yB]
    pure $ Complex (x, y)

part3 :: Complex -> Int
part3 (Complex (xA, yA)) = length . filter shouldBeEngraved $ do
  x <- [xA..xA+1000]
  y <- [yA..yA+1000]
  pure $ Complex (x, y)

-- >>> [0, 10..100]
-- [0,10,20,30,40,50,60,70,80,90,100]

main :: IO ()
main = do
  a <- readAEquals <$!> TextIO.getContents
  print $ part1 a
  print $ part2 a
  print $ part3 a

:::

My girlfriend is learning python, we are taking on the challenges together, today I may upload her solution:

::: spoiler python

A=[-3344,68783]
R = [0, 0]
B= [A[0]+1000, A[1]+1000]
pointsengraved = 0
cycleright = 0


for i in range(A[1], B[1]+1):
    for j in range(A[0], B[0]+1):
        for k in range(100):
            R = [int(R[0] * R[0] - R[1] * R[1]), int(R[0] * R[1] + R[1] * R[0])]
            R = [int(R[0] / 100000), int(R[1] / 100000)]
            R = [int(R[0] + j), int(R[1] + i)]
            if -1000000>R[0] or R[0]>1000000 or -1000000>R[1] or R[1]>1000000:
                #print(".", end="")
                break
            cycleright += 1
        if cycleright == 100:
            pointsengraved += 1
            #print("+", end="")
        cycleright = 0
        R = [0, 0]
    #print()

print(pointsengraved)

:::

The commented out print statements produce an ascii map of the set, which can be cool to view at the right font size.

I was stuck for a while on part 2. I thought I was running into precision errors, so I re-implemented everything using exact integers, then I re-implemented it in python, but it turns out that I just had an off-by-1 error. It just so happened that my off-by-1 algorithm gave the correct solution to the sample input. Part 3 takes a while to compute.

Scheme/Guile

(use-modules (ice-9 rdelim))
(use-modules (ice-9 format))
(use-modules (srfi srfi-1))

(define (parse-file file-name)
  (let* ((p (open-input-file file-name))
        (equality-split (string-split (read-line p) #\=))
        (complex-str (list-ref equality-split 1))
        (complex-parts (map string->number (string-split
          (substring/read-only complex-str 1 (- (string-length complex-str) 1))
          #\,))))
    (+ (car complex-parts) (* 0+1i (list-ref complex-parts 1)))))
(define (special-div numerator divisor)
  (+
    (truncate/ (real-part numerator) (real-part divisor))
    (* 0+1i (truncate/ (imag-part numerator) (imag-part divisor)))))

(let* ((A (parse-file "notes/everybody_codes_e2025_q02_p1.txt")))
  (format #t "Input is ~a\n" A)
  (let loop ((R 0+0i) (i 3))
    (if (> i 0)
      (loop (+ (special-div (* R R) 10+10i) A) (- i 1))
      (format #t "P1 Answer: [~d,~a]\n\n"
              (inexact->exact (real-part R))
              (inexact->exact (imag-part R))))))


(define (check-engrave-range value)
  (and-map (lambda (x) (<= (abs x) 1000000)) (list (real-part value) (imag-part value))))
(define (check-engrave point)
  (let loop ((result 0+0i) (i 100))
    (if (and (> i 0) (check-engrave-range result))
      (loop (+ point (special-div (* result result) 100000+100000i)) (- i 1))
      (check-engrave-range result))))
(let* ((origin (parse-file "notes/everybody_codes_e2025_q02_p2.txt"))
       (engrave-count 0))
  (format #t "Input is ~a\n" origin)
  (let loop ((i 0))
    (if (<= i 1000) (begin
      (let loop ((j 0))
        (if (<= j 1000) (begin
          (set! engrave-count (+ engrave-count
                                 (if (check-engrave (+ origin (+ i (* j 0+1i)))) 1 0)))
          (loop (+ j 10)))))
      (loop (+ i 10)))))
  (format #t "P2 answer: ~a\n\n" engrave-count))


(let* ((origin (parse-file "notes/everybody_codes_e2025_q02_p3.txt"))
       (engrave-count 0))
  (format #t "Input is ~a\n" origin)
  (let loop ((i 0))
    (if (<= i 1000) (begin
      (let loop ((j 0))
        (if (<= j 1000) (begin
          (set! engrave-count
                (+ engrave-count (if (check-engrave (+ origin (+ i (* j 0+1i)))) 1 0)))
          (loop (+ j 1)))))
      (loop (+ i 1)))))
  (format #t "P3 answer: ~a\n\n" engrave-count))

FSharp
(my first submission failed, so I may not be as detailed here)

I laugh at how long this took me. At first I got stuck due to checked arithmetic. In FSharp you open the Checked module and you get the overloaded operators for primitives, whereas CSharp is checked {a + b}. I also mistakenly used distinct on part 3 when I didn't need it.

I did appreciate the choose functions of functional programming, which returns all of the Some values of a sequence of options. The PSeq library let me use 70% of the cpu on the calculations :)

module EveryBodyCodes2025.Quest02  

open System.IO  
open System.Text.RegularExpressions  
open FSharp.Collections.ParallelSeq  
open Checked  

[<Struct>]  
type ComplexNumber = 
    {X: int64; Y: int64}  
    static member (+) (a: ComplexNumber, b: ComplexNumber) = {X = a.X + b.X; Y = a.Y + b.Y }  
    static member (*) (a: ComplexNumber, b: ComplexNumber) = { X = ( a.X * b.X ) - (a.Y * b.Y); Y = (a.X * b.Y) + (a.Y * b.X) }  
    static member (/) (a: ComplexNumber, b: ComplexNumber) = { X = a.X / b.X; Y = a.Y / b.Y }  
    override this.ToString() = $"[{this.X},{this.Y}]"  
    
let parseInput file =  
    File.ReadAllLines file  
    |> fun lines ->  
        let reg = Regex(@"A=\[(-?\d+),(-?\d+)\]")  
        let res = reg.Match(lines[0])  
        match res.Success with  
        | true ->  
            let x = int64 res.Groups[1].Value  
            let y = int64 res.Groups[2].Value  
            {X = x; Y = y}  
        | false -> failwith "failed to parse"  
        
    
let part1 () =  
    parseInput "Inputs/Quest02/Q02_P01.txt"  
    |> fun a ->  
        [1..3]  
        |> List.fold (fun acc _ -> (acc * acc) / { X = 10; Y = 10 } + a ) {X = 0; Y = 0}  

let cycle p =  
    let rec iterate current iteration =  
        if iteration = 100 then Some current  
        else  
            let next = (current * current) / {X = 100_000; Y = 100_000 } + p  
            if abs(next.X) > 1_000_000 || abs(next.Y) > 1_000_000 then  
                None  
            else iterate next (iteration + 1)  
    iterate { X = 0; Y = 0 } 0  

let part2 () =  
    parseInput "Inputs/Quest02/Q02_P02.txt"  
    |> fun a ->  
        seq {  
            for y in 0..100 do  
            for x in 0..100 do  
                yield {X = a.X + (int64 x * 10L); Y = a.Y + (int64 y * 10L)}  
        }  
        |> PSeq.choose cycle  
        |> PSeq.distinct  
        |> PSeq.length  

let part3 () =  
    parseInput "Inputs/Quest02/Q02_P03.txt"  
    |> fun a ->  
        seq {  
            for y in 0..1000 do  
            for x in 0..1000 do  
                yield {X = a.X + (int64 x); Y = a.Y + (int64 y)}  
        }  
        |> PSeq.choose cycle  
        |> PSeq.length  

For quest 2, I decided to implement my own Complex type and operators, because I expected parts 2 and 3 to have something unconventional, but alas it's just a regular Mandelbrot fractal.

Nim again, Nim forever:

type Complex = tuple[x,y: int]
proc `$`(c: Complex): string = &"[{c.x},{c.y}]"
proc `+`(a,b: Complex): Complex = (a.x+b.x, a.y+b.y)
proc `-`(a,b: Complex): Complex = (a.x-b.x, a.y-b.y)
proc `*`(a,b: Complex): Complex = (a.x*b.x - a.y*b.y, a.x*b.y + a.y*b.x)
proc `/`(a,b: Complex): Complex = (a.x div b.x, a.y div b.y)
proc `/`(a: Complex, b: int): Complex = (a.x div b, a.y div b)

proc parseInput(input: string): Complex =
  let parts = input.split({'=','[',',',']'})
  (parseInt(parts[2]), parseInt(parts[3]))

proc isStable(point: Complex, iter: int): bool =
  var num: Complex
  for _ in 1..iter:
    num = (num * num) / 100_000 + point
    if num.x notin -1_000_000 .. 1_000_000 or
       num.y notin -1_000_000 .. 1_000_000:
      return false
  true

proc solve_part1*(input: string): Solution =
  let start = parseInput(input)
  var point: Complex
  for _ in 1..3:
    point = (point * point) / 10 + start
  result := $point

proc solve_part2*(input: string): Solution =
  let start = parseInput(input)
  for y in 0..100:
    for x in 0..100:
      let point: Complex = (start.x + 10 * x, start.y + 10 * y)
      if point.isStable(iter=100): inc result.intVal

proc solve_part3*(input: string): Solution =
  let start = parseInput(input)
  for y in 0..1000:
    for x in 0..1000:
      let point: Complex = (start.x + x, start.y + y)
      if point.isStable(iter=100): inc result.intVal

Full solution at Codeberg: solution.nim

Rust

use log::debug;
use std::collections::HashSet;

use regex::Regex;

#[derive(PartialEq, Eq, Hash, Clone)]
struct Number(isize, isize);

impl Number {

  fn add(self: &Number, b: &Number) -> Number {
    Number(self.0 + b.0, self.1 + b.1)
  }

  fn mul(self: &Number, b: &Number) -> Number {
    Number(self.0 * b.0 - self.1 * b.1, self.0 * b.1 + self.1 * b.0)
  }

  fn div(self: &Number, b: &Number) -> Number {
    Number(self.0 / b.0, self.1 / b.1)
  }
}

pub fn solve_part_1(input: &str) -> String {
  let re = Regex::new(r"A=\[(\d+),(\d+)\]").unwrap();
  let (_, [x, y]) = re.captures(input).unwrap().extract();
  let a = Number(x.parse().unwrap(), y.parse().unwrap());
  let mut res = Number(0, 0);
  for _ in 0..3 {
    res = res.mul(&res);
    res = res.div(&Number(10, 10));
    res = res.add(&a);
  }
  format!("[{},{}]", res.0, res.1)
}

pub fn solve_part_2(input: &str) -> String {
  let re = Regex::new(r"A=\[([-0-9]+),([-0-9]+)\]").unwrap();
  let (_, [x, y]) = re.captures(input).unwrap().extract();
  let a = Number(x.parse().unwrap(), y.parse().unwrap());
  let mut engraved_points = 0;
  let mut pts: HashSet<_> = HashSet::new();
  for i in 0..=100 {
    for j in 0..=100 {
      let pt = Number(a.0 + 10 * i, a.1 + 10 * j);
      let mut res = Number(0, 0);
      engraved_points += 1;
      pts.insert(pt.clone());
      for _ in 0..100 {
        res = res.mul(&res);
        res = res.div(&Number(100_000, 100_000));
        res = res.add(&pt);
        if res.0.abs() > 1_000_000 || res.1.abs() > 1_000_000 {
          engraved_points -= 1;
          pts.remove(&pt);
          break;
        }
      }
    }
  }
  for i in 0..=100 {
    debug!("{}", (0..=100).map(|j| if pts.contains(&Number(a.0 + 10*i, a.1 + 10*j)) { 'X' } else {'.'}).collect::<String>());
  }
  engraved_points.to_string()
}

pub fn solve_part_3(input: &str) -> String {
  let re = Regex::new(r"A=\[([-0-9]+),([-0-9]+)\]").unwrap();
  let (_, [x, y]) = re.captures(input).unwrap().extract();
  let a = Number(x.parse().unwrap(), y.parse().unwrap());
  let mut engraved_points = 0;
  for i in 0..=1000 {
    for j in 0..=1000 {
      let pt = Number(a.0 + i, a.1 + j);
      let mut res = Number(0, 0);
      engraved_points += 1;
      for _ in 0..100 {
        res = res.mul(&res);
        res = res.div(&Number(100_000, 100_000));
        res = res.add(&pt);
        if res.0.abs() > 1_000_000 || res.1.abs() > 1_000_000 {
          engraved_points -= 1;
          break;
        }
      }
    }
  }
  engraved_points.to_string()
}

Sadly Uiua doesn't handle the large numbers here well, so here's a Dart solution instead.

import 'dart:math';
import 'package:more/more.dart';

typedef P = Point;
void main(List<String> args) {
  P add(P a, P b) => P(a.x + b.x, a.y + b.y);
  P mul(P a, P b) => P(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x);
  P div(P a, P b) => P(a.x ~/ b.x, a.y ~/ b.y);
  P part1(P a) {
    P r = P(0, 0);
    for (var _ in 0.to(3)) {
      r = add(div(mul(r, r), P(10, 10)), a);
    }
    return r;
  }

  bool test2(P a) {
    P r = P(0, 0);
    for (var _ in 0.to(100)) {
      r = add(div(mul(r, r), P(100000, 100000)), a);
      if (r.x.abs() > 1000000 || r.y.abs() > 1000000) return false;
    }
    return true;
  }

  part2(P a, {step = 1}) => [
        for (var x in 0.to(1001, step: step))
          for (var y in 0.to(1001, step: step)) test2(P(a.x + x, a.y + y))
      ].count((e) => e);

  print(part1(P(25, 9)));
  print(part2(P(35300, -64910), step: 10));
  print(part2(P(35300, -64910)));
}