Python

# pairwise helps picking consecutive values easier
#   [A,B,C,D] -> [AB, BC, CD]
# combinations will give me combinations of elements in a sequence
#   [A,B,C,D] -> [AB, AC, AD, BC, BD, CD]
from itertools import pairwise, combinations


# line will pass thorough the center if the points are on opposite ends,
#   i.e., total points / 2 distance away
def part1(data: str, nail_count: int = 32):
    opposite_dist = nail_count // 2
    nodes = [int(n) for n in data.split(",")]

    center_passes = 0
    for a, b in pairwise(nodes):
        if abs(a - b) == opposite_dist:
            center_passes += 1
    return center_passes


assert part1("1,5,2,6,8,4,1,7,3", 8) == 4


# BRUTEFORCE: take every two points and check if they intersect
def part2(data: str, nail_count: int = 32):
    def intersects(s1, s2):
        # expand the points
        (x1, x2), (y1, y2) = s1, s2
        # make sure x1 is the smaller nail and x2 is the bigger one
        if x1 > x2:
            x1, x2 = x2, x1
        # there is an intersection if one point lies bwtween x1 and x2
        #  and the other lies outside it
        if x1 < y1 < x2 and (y2 > x2 or y2 < x1):
            return True
        if x1 < y2 < x2 and (y1 > x2 or y1 < x1):
            return True
        return False

    nodes = [int(n) for n in data.split(",")]
    knots = 0
    for s1, s2 in combinations(pairwise(nodes), 2):
        if intersects(s1, s2):
            knots += 1

    return knots


assert part2("1,5,2,6,8,4,1,7,3,5,7,8,2", 8) == 21

from collections import defaultdict


# This is better than bruteforce
def part3(data: str, nail_count: int = 256):
    connections = defaultdict(list)
    nodes = [int(n) for n in data.split(",")]

    # record all connections, both ways
    for a, b in pairwise(nodes):
        connections[a].append(b)
        connections[b].append(a)

    max_cuts = 0
    for node_a in range(1, nail_count + 1):
        cuts = 0
        for node_b in range(node_a + 2, nail_count + 1):
            # add new cuts for the node we just added between a and b
            for other_node in connections[node_b - 1]:
                if other_node > node_b or other_node < node_a:
                    cuts += 1
            # also add any cuts for threads that go between node a and b
            cuts += connections[node_a].count(node_b)

            # remove cuts that were going from BETWEEN node_a and node_b-1 (prev node_b) to node_b
            for other_node in connections[node_b]:
                if node_a < other_node < node_b:
                    cuts -= 1
            # also remove any cuts for threads that go between node a and b-1
            cuts -= connections[node_a].count(node_b - 1)

            max_cuts = max(max_cuts, cuts)

    return max_cuts


assert part3("1,5,2,6,8,4,1,7,3,6", 8) == 7

Rust

pub fn solve_part_1(input: &str) -> String {
    let numbers: Vec<i32> = input.split(",").map(|x| x.parse().unwrap()).collect();
    let mut count = 0;
    for i in 1..numbers.len() {
        if numbers[i].abs_diff(numbers[i - 1]) == 16 {
            count += 1;
        }
    }
    count.to_string()
}

pub fn solve_part_2(input: &str) -> String {
    let numbers: Vec<i32> = input.split(",").map(|x| x.parse().unwrap()).collect();
    let mut lines: Vec<(i32, i32)> = vec![];
    for i in 1..numbers.len() {
        let (a, b) = (numbers[i - 1], numbers[i]);
        if a > b {
            lines.push((b, a));
        } else {
            lines.push((a, b));
        }
    }
    let mut knots = 0;
    for i in 0..lines.len() {
        for j in 0..i {
            let (a, b) = lines[i];
            let (c, d) = lines[j];
            if a == c || a == d || b == c || b == d {
                continue;
            }
            let c_inside = c > a && c < b;
            let d_inside = d > a && d < b;
            if c_inside != d_inside {
                knots += 1;
            }
        }
    }
    knots.to_string()
}

pub fn solve_part_3(input: &str) -> String {
    let numbers: Vec<i32> = input.split(",").map(|x| x.parse().unwrap()).collect();
    let mut lines: Vec<(i32, i32)> = vec![];
    for i in 1..numbers.len() {
        let (a, b) = (numbers[i - 1], numbers[i]);
        if a > b {
            lines.push((b, a));
        } else {
            lines.push((a, b));
        }
    }
    let mut best_cut_threads = i64::MIN;
    for d in 1..=256 {
        for c in 1..d {
            let mut cut_threads = 0;
            for (a, b) in lines.iter().copied() {
                if a == c || a == d || b == c || b == d {
                    if a == c && b == d {
                        cut_threads += 1;
                    }
                    continue;
                }
                let c_inside = c > a && c < b;
                let d_inside = d > a && d < b;
                if c_inside != d_inside {
                    cut_threads += 1;
                }
            }
            if cut_threads > best_cut_threads {
                best_cut_threads = cut_threads;
            }
        }
    }
    best_cut_threads.to_string()
}

Haskell

Woo! I got on the leaderboard at last. I don't think I've seen a problem like this one before, but fortunately it wasn't as tricky as it seemed at first glance.

import Control.Monad  
import Data.List  
import Data.List.Split  
import Data.Tuple  

readInput :: String -> [(Int, Int)]  
readInput = map fixOrder . (zip <*> tail) . map read . splitOn ","  
  where  
    fixOrder (x, y)  
      | x > y = (y, x)  
      | otherwise = (x, y)  

crosses (a, b) (c, d) =  
  not (a == c || a == d || b == c || b == d)  
    && ((a < c && c < b) /= (a < d && d < b))  

part1 n = length . filter ((== n `quot` 2) . uncurry (-) . swap)  

part2 n = sum . (zipWith countKnots <*> inits)  
  where  
    countKnots x strings = length $ filter (crosses x) strings  

part3 n strings =  
  maximum [countCuts (a, b) | a <- [1 .. n - 1], b <- [a + 1 .. n]]  
  where  
    countCuts x = length $ filter (\s -> x == s || x `crosses` s) strings  

main =  
  forM_  
    [ ("everybody_codes_e2025_q08_p1.txt", part1 32),  
      ("everybody_codes_e2025_q08_p2.txt", part2 256),  
      ("everybody_codes_e2025_q08_p3.txt", part3 256)  
    ]  
    $ \(input, solve) -> readFile input >>= print . solve . readInput  

Nim

Part 2 - I just really didn't want to think that day. So when puzzle asked me to check if lines intersect - I wrote the intersection checking solution with 2D points.

Part 3 is geometry + bruteforce.

proc solve_part1*(input: string): Solution =
  let pins = input.split(',').mapIt(parseInt(it))
  for i in 0 ..< pins.high:
    let d = abs(pins[i] - pins[i+1])
    if d == 16: inc result.intVal

proc ccw(A,B,C: Vec2): bool = (C.y-A.y) * (B.x-A.x) > (B.y-A.y) * (C.x-A.x)
proc isIntersection(A,B,C,D: Vec2): bool = ccw(A,C,D) != ccw(B,C,D) and ccw(A,B,C) != ccw(A,B,D)

proc solve_part2*(input: string): Solution =
  const two_pi = PI * 2
  const pin_count = 256

  var pins: array[pin_count, Vec2]
  for i in 0 ..< pin_count:
    let angle = two_pi * (i / pin_count)
    let point: Vec2 = (cos(angle), sin(angle))
    pins[i] = point

  let inst = input.split(',').mapIt(parseInt(it))
  var lines: seq[(Vec2, Vec2)]

  for i in 0 ..< inst.high:
    let A = pins[inst[i]-1]
    let B = pins[inst[i+1]-1]

    for (C, D) in lines:
      if isIntersection(A,B,C,D):
        inc result.intVal
    lines.add shortenSegment(A, B, 0.0001)

proc solve_part3*(input: string): Solution =
  const two_pi = PI * 2
  const pin_count = 256

  var pins: array[pin_count, Vec2]
  for i in 0 ..< pin_count:
    let angle = two_pi * (i / pin_count)
    let point: Vec2 = (cos(angle), sin(angle))
    pins[i] = point

  let inst = input.split(',').mapIt(parseInt(it))
  var lines: seq[(Vec2, Vec2)]

  for i in 0 ..< inst.high:
    let A = pins[inst[i]-1]
    let B = pins[inst[i+1]-1]
    lines.add shortenSegment(A, B, 0.0001)

  var bestSum = 0
  for i in 0 ..< pin_count:
    for j in i+1 ..< pin_count:
      let A = pins[i]
      let B = pins[j]
      var sum = 0
      for (C, D) in lines:
        if isIntersection(A,B,C,D): inc sum
      if sum > bestSum: bestSum = sum
  result := bestSum

Full solution at Codeberg: solution.nim

Common Lisp's loop macro has a pretty crazy list of features. New ones in this solution are for x on xs, which binds the iteration variable x to successive tails (instead of elements) of xs, and maximize, which is an alternative accumulator to sum. There's nothing interesting about the solution itself though -- simple brute-force enumeration.

(ql:quickload :str)

(defun parse-line (line)
  (mapcar #'parse-integer (str:split "," line)))

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

(defun pairs (ns)
  (loop for tail on ns
        if (not (null (cdr tail)))
          collect (cons (car tail) (cadr tail))))

(defun through-center? (nails segment)
  (destructuring-bind (x . y) segment
    (= (mod (- x y) nails) (/ nails 2))))

(defun main-1 (filename)
  (let ((positions (read-inputs filename)))
    (loop for segment in (pairs positions)
          sum (if (through-center? 32 segment) 1 0))))

(defun crosses? (seg1 seg2)
  "When everything is normalized to 1..nails indices, seg1 crosses seg2 iff one of seg2's
  endpoints lies strictly between the endpoints of seg1, and the other one of seg2's endpoints
  lies strictly below or above the endpoints of seg1."
  (destructuring-bind (x1 . y1) seg1
    (destructuring-bind (x2 . y2) seg2
      (let ((big1 (max x1 y1))
            (small1 (min x1 y1)))
        (or (and (< small1 x2 big1)
                 (or (< y2 small1) (> y2 big1)))
            (and (or (< x2 small1) (> x2 big1))
                 (< small1 y2 big1)))))))

(defun main-2 (filename)
  (let ((positions (read-inputs filename)))
    (loop for seg-list on (pairs positions)
          sum (loop for seg2 in (cdr seg-list)
                    sum (if (crosses? (car seg-list) seg2) 1 0)))))

(defun score (threads seg)
  (loop for thread in threads
        sum (if (crosses? thread seg) 1 0)))

(defun main-3 (filename)
  (let* ((positions (read-inputs filename))
         (threads (pairs positions))
         (nails 256))
    (loop for x1 from 1 to nails
          maximize (loop for y1 from (1+ x1) to nails
                         maximize (score threads (cons x1 y1))))))

Uiua

Just a dirty great hack and a few minutes of toasty CPU for part3 with live data.

"1,5,2,6,8,4,1,7,3"
⊜⋕⊸≠@,
&p /+=⊃(⧈₂(⌵/-)|÷2/↥) # Part1 --> 4

"1,5,2,6,8,4,1,7,3,5,7,8,2"
⊜⋕⊸≠@,
Knot  ← (⊃(=¯|=)∩⌞(±-)⊙°⊟)
Knots ← /↧/↥↯∞_2[∩⌟Knot]°⊟
⧈₂⍆
&p /+≡(/+≡Knots¤°⊂↙¯)⊙¤+1↘2⇡⊸⧻ # Part 2 --> 21

"1,5,2,6,8,4,1,7,3,6"
⊜⋕⊸≠@,
⊃(⧅<2+1⇡⧻◴|⧈₂⍆)            # Possible cuts, existing strings.
/↥+⊃(≡˜∊⊙¤|≡(/+≡Knots¤)⊙¤) # Part3 --> 7

Scheme/Guile

Takes about 5 seconds.

(import (rnrs io ports (6))
        (srfi srfi-1))
#!curly-infix

(define (parse-file file-name)
  (let ((sequence (map string->number (string-split (string-trim-both (call-with-input-file file-name get-string-all)) #\,))))
    (zip sequence (cdr sequence))))

(let loop ((sequence (parse-file "notes/everybody_codes_e2025_q08_p1.txt")) (count 0))
  (if (null? sequence)
      (format #t "P1 Answer: ~a\n\n" count)
      (loop (cdr sequence) (+ count (if (and last (eq? (modulo (- (cadar sequence) (caar sequence)) 32) 16)) 1 0)))))

(define (crosses-over? a b)
  (let ((a1 (car a))
        (a2 (cadr a))
        (b1 (car b))
        (b2 (cadr b)))
    (let ((a2 (modulo {a2 - a1} 256))
          (b1 (modulo {b1 - a1} 256))
          (b2 (modulo {b2 - a1} 256)))
      (and (not (eq? b1 0)) (not (eq? b2 0))
      (or
        (and {b1 < a2} {b2 > a2})
        (and {b1 > a2} {b2 < a2}))))))
(define (count-cross-overs sequence a)
  (let loop ((sequence sequence) (count 0))
    (if (null? sequence)
        count
        (loop (cdr sequence) (+ count (if (crosses-over? (car sequence) a) 1 0))))))
(let loop ((sequence (parse-file "notes/everybody_codes_e2025_q08_p2.txt")) (passed '()) (count 0))
  (if (null? sequence)
      (format #t "P2 Answer: ~a\n\n" count)
      (loop (cdr sequence) (cons (car sequence) passed) (+ count (count-cross-overs passed (car sequence))))))


(let ((sequence (parse-file "notes/everybody_codes_e2025_q08_p3.txt")))
  (let loop ((i 1) (greatest 0))
    (if {i > 256}
        (format #t "P3 Answer: ~a\n\n" greatest)
        (loop (1+ i) (max greatest (let loop ((j i) (greatest 0))
              (if {j > 256}
                  greatest
                  (loop (1+ j) (max greatest (count-cross-overs sequence (list i j)))))))))))