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  

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()
}

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

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)

{-# 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:

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  

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)));
}

I was sad to discover that division wasn't actual complex division. As in some other languages, one has to implement this "division" by hand because the library designers assume that you never want truncating integer division together with complex numbers. Which is reasonable, I guess.

(ql:quickload :cl-ppcre)
(ql:quickload :str)

(defun parse-line (line)
  (ppcre:register-groups-bind
      (name x y)
      ("^([A-Za-z]+)=[[]([\-0-9.]+),([\-0-9.]+)[]]$" line)
    (cons name (complex (parse-integer x) (parse-integer y)))))

(defun read-inputs (filename)
  (let ((input-lines (uiop:read-file-lines filename)))
    (parse-line (car input-lines))))

(defun fake-complex-division (y d)
  (complex (truncate (realpart y) (realpart d))
           (truncate (imagpart y) (imagpart d))))

(defun f (x a)
  (+ a (fake-complex-division (* x x) #C(10 10))))

(defun main-1 (filename)
  (let* ((a (cdr (read-inputs filename)))
         (result (f (f (f #C(0 0) a) a) a)))
    (str:concat "["
                (write-to-string (realpart result))
                ","
                (write-to-string (imagpart result))
                "]")))

(defun engrave? (p)
  (labels ((iter (x n)
             (if (zerop n)
                 t
                 (let ((next-x (+ p (fake-complex-division (* x x) #C(100000 100000)))))
                   (if (or (> (abs (realpart next-x)) 1000000)
                           (> (abs (imagpart next-x)) 1000000))
                       nil
                       (iter next-x (- n 1)))))))
    (iter 0 100)))

(defun count-engraves (a spacing)
  (let ((steps (/ 1000 spacing)))
    (loop for x from 0 to steps
          sum (loop for y from 0 to steps
                    for p = (+ a (complex (* spacing x) (* spacing y)))
                    sum (if (engrave? p) 1 0)))))

(defun main-2 (filename)
  (count-engraves (cdr (read-inputs filename)) 10))

(defun main-3 (filename)
  (count-engraves (cdr (read-inputs filename)) 1))