I was playing Chutes & Ladders with my four-year-old daughter yesterday, and I thought, “How long is this going to take?”

I saw an interesting mathematical analysis of the game a few years ago, but it seems to be offline, though you can read it via the wayback machine.

But that didn’t answer my specific question, namely, “How long is this going to take?”

So I wrote a bit of R code to simulate the game.

Here’s the distribution of the number of spins to complete the game, by number of players:

With two players, the average number of spins is 52, with a 90th percentile of 88.

If you add a third player, the average increases to 65, and the 90th percentile increases to 103. You’re playing fewer rounds, but each round is three times as long. If you add a fourth player, the average is 76 and the 90th percentile is 117.

So, in trying to minimize the agony, it seems best to not encourage my eight-year-old son to join us in the game. If he plays with us, there’s a 63% chance that it will take longer.

And that’s particularly true because then the chance of my daughter winning drops from about 1/2 to about 1/3.

That raises another question: if I let her go first, what advantage does that give her? Not much. The chance that the person who goes first will win is 50.9%, 34.4%, and 25.9%, respectively, when there are 2, 3, and 4 players. So not a noticeable amount. Thus I cheat (on her behalf). Really, though, I’m cheating in order to shorten the game as much as to ensure that she wins.

*Note*: There’s a close connection between this problem and my work on the multiple-strain recombinant inbred lines. (See this and that.) I’m tempted to play around with it some more.

Additional numerical results here.

Tags: kids, Markov chains, simulation

18 May 2013 at 1:23 am |

You know that your daughter will find this post within the next 5-10 years and will not be happy, right?

18 May 2013 at 5:56 am |

I think she’ll understand.

18 May 2013 at 2:28 am |

Nice ! I did publish also a post a long time ago on that game too, http://freakonometrics.hypotheses.org/2327 with R codes, and another one (in French) http://freakonometrics.hypotheses.org/1296 on “what should it be optimal to get with your dice to win faster, on average”…

20 May 2013 at 11:34 am |

I wrote even more about the game.

18 May 2013 at 10:40 am |

Cool. I also did a snakes and latters post (after seeing Arthur’s ;)

Mine looks at entropy as a measure of prediction uncertainty.

http://bayesianbiologist.com/2011/12/31/uncertainty-in-markov-chains-fun-with-snakes-and-ladders/

20 May 2013 at 11:34 am |

The analysis of the game is remarkably interesting, especially given how boring it is to play. :)

28 May 2013 at 4:12 am |

[…] couple of days ago, we had a quick chat onĀ Karl Broman‘s blog, about snakes and ladders (see https://kbroman.wordpress.com/… ;) with Karl and Corey (see http://bayesianbiologist.com/….), and the use of Markov Chain. I do […]