Chutes And Ladders
March 4, 2011
Our method is to simulate playing a large number of games and count the outcomes. Chutes and ladders are represented as association-lists with the from square in the car and to square in the cdr:
(define chutes '((16 . 6) (47 . 26) (49 . 11)
(56 . 53) (62 . 19) (64 . 60) (87 . 24)
(93 . 73) (95 . 75) (98 . 78)))
(define ladders '((1 . 38) (4 . 14) (9 . 31)
(21 . 42) (28 . 84) (36 . 44) (51 . 67)
(71 . 91) (80 . 100)))
A single game is the path that one token takes to reach space 100. The first cond clause ends the game, the second cond clause stays if the roll of the die is beyond the end, the third cond clause handles chutes and ladders, and the final cond clause handles all other squares. The loop accumulates the path through the board; p is the current position, and ps is the path. The lambda in the third clause takes the “true” result of an a-list lookup, which is a start/end pair, and extracts the ending position:
(define (game)
(let loop ((ps '(0)))
(let* ((die (randint 6 0)) (p (+ (car ps) die)))
(cond ((= 100 (car ps)) (cdr (reverse ps)))
((< 100 p) (loop (cons (car ps) ps)))
((or (assoc p chutes) (assoc p ladders))
=> (lambda (x) (loop (cons (cdr x) ps))))
(else (loop (cons p ps)))))))
For instance, a sample game might look like this; the player hit the 9/31 ladder and the 80/100 ladder, but no chutes, for a 16-roll game:
> (game)
(6 7 31 37 42 48 50 55 57 63 65 68 72 74 79 100)
Function games executes (game) multiple times and reports the lengths of the various games played:
(define (games n)
(let loop ((n n) (gs '()))
(if (zero? n) gs
(loop (- n 1) (cons (length (game)) gs)))))
Compete reports the lengths of n games each played by k players. The expression (apply min (games k)) reports the number of turns taken by the winner:
(define (compete k n)
(let loop ((n n) (gs '()))
(if (zero? n) gs
(loop (- n 1) (cons (apply min (games k)) gs)))))
Finally, stats gathers statistics:
(define (stats k n)
(let ((gs (compete k n)))
(values (apply min gs) (apply max gs)
(exact->inexact (/ (apply + gs) n)))))
Here are some sample results:
> (stats 1 1000000)
7
349
39.290771
> (stats 2 100000)
7
154
26.3613
> (stats 3 100000)
7
111
21.73284
> (stats 4 100000)
7
94
19.20292
> (stats 5 100000)
7
80
17.70565
The minimum number of turns to complete the game is 7 (Wikipedia says 6; perhaps their board is somewhat different than ours). It takes on average 39.3 turns to reach space 100. Games get shorter as the number of players increases, since at least one of the players beats the average. The poor fellow who rolled 349 times before reaching space 100 was terribly unlucky; presumably, he should wait until tomorrow to buy a lottery ticket, as he certainly has no luck today.
We used randint from the Standard Prelude. You can run the program at http://programmingpraxis.codepad.org/RPq4Ocqx.
My Haskell solution (see http://bonsaicode.wordpress.com/2011/03/04/programming-praxis-chutes-and-ladders/ for a version with comments):
import Control.Monad import System.Random promote :: Num a => a -> a promote n = maybe n id $ lookup n [(16,6), (47,26), (49,11), (56,53), (62,19), (64,60), (87,24), (93,73), (95,75), (98,78), (1,38), (4,14), (9,31), (21,42), (28,84), (36,44), (51,67), (71,91), (80,100)] game :: IO [Int] game = fmap (tail . turn 0 . randomRs (1,6)) newStdGen where turn 100 _ = [100] turn n ~(d:ds) = n : turn (if n+d > 100 then n else promote $ n+d) ds stats :: Int -> Int -> IO (Int, Int, Float) stats k n = fmap (\rs -> (minimum rs, maximum rs, average rs)) . replicateM n . fmap minimum . replicateM k $ fmap length game where average xs = fromIntegral (sum xs) / fromIntegral n main :: IO () main = do print =<< stats 1 100000 print =<< stats 2 100000 print =<< stats 3 100000 print =<< stats 4 100000 print =<< stats 5 100000[…] today’s Programming Praxis exercise, our goal is to simulate the board game Chutes and Ladders. Let’s […]
Not terribly fast, though it gets the job done. To speed this up, I tried
1. Numpy arrays (which doesn’t get much of an improvement),
2. PyPy (which sped things up a lot), and
3. Cython to compile the code.
None of these three options come standard with the typical Python installation,
unfortunately. As a further exercise for myself, I went about compiling via
Cython, without changing the original code beyond all recognition; details
available on codepad.
#!/usr/bin/env python from random import randrange def single_game(chutes, ladders): pos, path = 0, [] while pos < 100: pos += randrange(1, 7) if pos in chutes: pos = chutes[pos] elif pos in ladders: pos = ladders[pos] path.append(pos) if path[-1] > 100: path[-1] = 100 return path def mult_games(n, chutes, ladders): return [len(single_game(chutes, ladders)) for _ in xrange(n)] def compete(k, n, chutes, ladders): return [min(mult_games(k, chutes, ladders)) for _ in xrange(n)] def mean(xs): return sum(xs) / float(len(xs)) def stats(k, n, chutes, ladders): games = compete(k, n, chutes, ladders) return min(games), max(games), mean(games) if __name__ == "__main__": chutes = {16: 6, 47: 26, 49: 11, 56: 53, 62: 19, 64: 60, 87: 24, 93: 73, 95: 75, 98: 78} ladders = {1: 38, 4: 14, 9: 31, 21: 42, 28: 84, 36: 44, 51: 67, 71: 91, 80: 100} for k in xrange(1, 7): print stats(k, 100000, chutes, ladders)I’ve got a Haskell solution similar to Remco Niemeijer’s one, so I decided not to repeat it here once more.
Instead, I solved the first question regarding the minimal number of rolls in a “theoretical” way, without simulating the game itself. I converted the game board into a directed graph — each node has 6 outgoing edges, corresponding to 6 possible rolls (or less, if it’s near 100) and then used breadth-first search to find out the shortest path length from start to finish. It turned out to be 7.
import Data.Graph.Inductive hiding (nodes, edges, run) import Data.Graph.Inductive.Query.BFS (esp, lesp) start = 0 finish = 100 step = 6 transitions = [ (1, 38), (4, 14), (9, 31), (16, 6), (21, 42), (28, 84) , (36, 44), (47, 26), (49, 11), (51, 67), (56, 53), (62, 19) , (64, 60), (71, 91), (80, 100), (87, 24), (93, 73), (95, 75) , (98, 78) ] nodes :: [(Int, ())] nodes = [(n, ()) | n <- [start .. finish]] edges :: [(Int, Int, ())] edges = concatMap (edges' . fst) nodes where edges' x = [ edge x n (lookup n transitions) | k <- [1 .. step], let n = x + k, n <= finish ] edge x n (Just trans) = (x, trans, ()) edge x n Nothing = (x, n, ()) gameGraph :: Gr () () gameGraph = mkGraph nodes edges shortestGame :: Int shortestGame = length (esp start finish gameGraph) - 1Here’s my python solution. Uses the dijkstra function from a previous exercise to find the minimum path.
from dijkstra import * from random import randrange chutes = {98:78, 95:75, 93:73, 87:24, 64:60, 62:19, 56:53, 49:11, 47:26, 16: 6 } ladders = { 1:38, 4:14, 9:31, 21:42, 28:84, 36:44, 51:67, 71:91, 80:100 } special = dict(chutes.viewitems() | ladders.viewitems()) # transision table: next_sq[current square][dice roll - 1] -> next square # includes effects of slides and ladders, so next_sq[0][1] is 38 next_sq = [[special.get(r,min(r, 100)) for r in range(n+1,n+7)] for n in range(101)] # create graph from the transition table graph = {n:{t:1 for t in next_sq[n]} for n in range(101)} mst = dijkstra(graph,0,100) mst[100] # returns (7, 74), min of 7 turns, previous sequare is 74 def play(nplayers=1): sq = [0]*nplayers nturns = 0 while True: nturns += 1 for player in range(nplayers): sq[player] = next_sq[sq[player]][randrange(6)] if sq[player] == 100: return nturns, player for nplayers in range (1,6): turns = [play(nplayers)[0] for _ in range(10000)] print float(sum(turns))/len(turns), min(turns)These implementations which walk a tree take a long time to run – the first haskell program takes on the order of 30 seconds to print the first statistics on my system. A faster way is to create a transition matrix and repeatedly multiply with a state vector to compute the probability of any particular state after N turns. Experimentally, after 350 iterations, this accounts for 99.9999% of all games and produces an answer of avg turns = 39.2 in under a second of CPU time.
My Try in Rexx, Using the Dijkstra-Algorithm from https://programmingpraxis.com/2011/01/04/dijkstras-algorithm/ to find the optimal strategy
MAX = 99999 chutes = '16-6 47-26 49-11 56-53 64-60 62-19 87-24 93-73 95-75 98-78' ladders = '1-38 4-14 10-31 28-84 21-42 36-44 51-67 71-91 80-100' ch. = '' ld. = '' erg. = '' cnt = 0 orte = '' strecken = '' call init_basic call zufall 100, 1 call zufall 100, 3 call zufall 100, 10 call zufall 100, 100 call ergebnis_zufall start = 1 ziel = 100 d. = MAX d.start = 0 p. = '' call init_shortest_path call optimal_path exit zufall: parse arg t, p ct_sum = 0 do games = 1 to t hundert = 0 player_pos. = 0 do while hundert == 0 /* ---------------------- */ /* p = Anzahl der Spieler */ /* ---------------------- */ do player = 1 to p wrk = 0 cp = player_pos.player if cp == 100 then do hundert = 1 leave end select /* -------------------------------- */ /* Akt. Pos = Beginn einer Leiter ? */ /* Neue Position = Ende der Leiter */ /* -------------------------------- */ when ch.cp > 0 then wrk = ch.cp /* -------------------------------- */ /* Akt. Pos = Oben an Rutsche ? */ /* Neue Position = Unten an Rutsche */ /* -------------------------------- */ when ld.cp > 0 then wrk = ld.cp otherwise do /* -------------------------------- */ /* Wuerfeln Zufallszahl zw. 1 und 5 */ /* -------------------------------- */ wrk = cp + random(1, 6) ct_sum = ct_sum + 1 /* --------------------------------------*/ /* wenn neue Pos. > 100, dann ignorieren */ if wrk > 100 then iterate player /* --------------------------------------*/ end end player_pos.player = wrk end end end cnt = cnt + 1 erg.cnt = p 'Personen mussten (/)', 'bei' t 'Versuchen durchschn.', ((ct_sum % p) % t) 'mal wuerfeln.' return init_basic: do while words(chutes) > 0 parse value chutes with first chutes parse value first with start'-'end ch.start = end end do while words(ladders) > 0 parse value ladders with first ladders parse value first with start'-'end ld.start = end end return init_shortest_path: do x = 1 to 100 orte = orte x end do x = 1 to 100 ks = '' if ch.x > '' then iterate if ld.x > '' then ks = ks x'-'ld.x'-'0 do y = x+1 to min(x+6,100) if strip(ch.y) == '' then ks = ks x'-'y'-'1 end strecken = strecken ks end return ergebnis_zufall: say cnt do i = 1 to cnt say erg.i end return optimal_path: q = orte do while words(q) > 0 u = n_nb(q) if u == '' then leave q = delword(q, wordpos(u,q), 1) call relax u,strecken end say reverse(translate(strip(ausgabe(ziel, '')),'<',' ')), '=' d.ziel 'Wuerfe' return n_nb: procedure expose d. MAX parse arg q min = MAX rw = '' do i = 1 to words(q) w = word(q, i) if d.w < min then do min = d.w rw = w end end return rw relax: procedure expose d. p. parse arg akt,kanten do while length(kanten) > 0 parse value kanten with kante kanten parse value kante with von'-'nach'-'entf select when akt == von then nb = nach otherwise iterate end neu = d.akt + entf if d.nb > neu then do d.nb = neu p.nb = akt end end return ausgabe: parse arg vorg, route if p.vorg == '' then return strip(reverse(route vorg)) return ausgabe(p.vorg,route vorg)