Proposal

 Senior Project Proposal

Joey Cimochowski

12/12/12

      I.            Title of Project:

Chutes, Ladders, and Numbers: a Mathematical Analysis of the Game Chutes and Ladders

    II.            Statement of Purpose:

The overarching question I am trying to answer is how can the game of Chutes and Ladders be modeled mathematically, and how does the structure of the game influence this model? This includes examining the probability of a player winning the game in a certain number of moves, the influence of the board’s configuration on the outcome of the game (How does the probability change if the squares containing chutes and ladders are changed? Was it designed to minimize the number of moves on average required to finish a game?), and the influence of the dice on the game (How does the  probability of victory change if the dice were eight-sided as opposed to six-sided? What if the dice were weighted?).

   III.          Background: 

           Chutes and Ladders was one of my favorite childhood games, and understanding how it works after playing it for so many hours as a kid is almost like solving a decade-long riddle. Additionally, Chutes and Ladders is fairly simple to play and is entirely governed by chance, making it an ideal subject of research for an inexperienced mathematician like myself. One way to model the question mathematically is with graph theory, a topic I have been studying in Mrs. Bailey’s Math Investigations Club, and while I haven’t done any prior research on this topic specifically, I have gained some experience with solving other graph theory problems.

 IV.        Prior Research: 

          Some curious mathematicians have already tackled parts of my question, and while this gives  me a useful place to start, many of the other questions I posed remain unanswered. The most extensive research on the topic that I could find comes from an entry in The Mathematical Gazette ("How Long Is a Game of Snakes and Ladders?",The Mathematical Gazette, Vol. 77). This article looks at many moves long the average game of Chutes and Ladders is expected to be as well as how adding or removing a ladder or chute would impact this length. A second source I found was a page from BYU’s website (http://math.byu.edu/~jeffh/mathematics/games/chutes/chutes.html, by Jeffrey Humphreys) which listed the expected probabilities that a game would finish in certain numbers of moves and discussed the theory behind these predictions. A third source I found was an entry on a math-related website (http://www.datagenetics.com/blog/november12011/index.html) which tested the probability of the game finishing in a certain number of moves by running several trials on a computer simulation of the game and then compared these values to those predicted by a purely mathematical model. All three of these used “Markov Chains” to arrive at their conclusions.

   V.            Significance:

Many people jeer that math “has no practical use” and feel that once they’ve completed basic algebra they’ve learned all the math that could be useful to them in their daily lives. Building an extensive mathematical model of Chutes and Ladders, a game almost all Americans are familiar with, will help show how useful mathematics can be in practical situations. For instance, besides predicting outcomes in Chutes and Ladders, graph-theoretic models can also be used to predict how information can spread throughout social networking sites. This is an especially important message to convey to children, the target audience of games like Chutes and Ladders, and showing them that even board games can be modeled with math may give them a deeper interest in the subject.

 VI.            Description:

The final product I will produce will be a report documenting all of my findings, which will follow from research on each question I posed. I plan answering these problems through Markov Chains, which are used to model random events. Prior to answering these questions, I will write a computer program which can simulate games of Chutes and Ladders that will allow me to confirm the validity of my results.  I also will do library research to ensure that I’m knowledgeable enough to solve the problems I posed.

VII.            Methodology:

For the few weeks, I will rent out library books on Markov Chains, Graph Theory, and Stochastic Processes – three areas of study that will be relevant to my problem. Following that, I will make a computer program that “plays” Chutes and Ladders, which will allow me to gather large samples of data about the average number of turns it takes to complete a game. I will then build a Markov Chain model of Chutes and Ladders that will allow me to predict mathematically how many rounds it takes on average to complete a game of Chutes and Ladders, and I can use the results of my program and other findings from my sources to verify these results. I will then start considering whether the Milton Bradley version of the game was designed in any particular way to minimize the number of turns. For instance, does the order they were placed in somehow minimize the number of average turns per game? Is there anything special about having a spinner with six possible moves? After all, while the game should still want to balance risk and reward by having a roughly equal number of chutes and ladders, kids would lose interest if it dragged on for too long. Once I’ve reached a conclusion, I will start revisiting some of my previous findings and see whether there is a simpler approach to solving these problems, as my sources hinted that computing probability from Markov Chains requires a lot of computing power. I will then bring all of my results together and write a report explaining what I found.

VIII.            Problems:

Some people feel that there is a fundamental divide between the “math world” and the “real world” and may doubt that something as real as Chutes and Ladders can be figured out with numbers on paper. While such a deep misconception may not be easy to correct, I would encourage such opponents to play the game for themselves and see if my findings really were an accurate prediction of results. After all, that is what models are all about: predicting reality.