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Limit Cases And Strategy In Chutes and Ladders

Vincent Ciarcia, Erik Insko

Abstract

We analyze what happens to the average duration of a game of Chutes and Ladders as the probability of rolling $δ\in \{ 1,2,3,4,5,6\}$ approaches 100%. We utilize Markov models, and Monte Carlo simulations in Python. We also introduce strategy to the board game by allowing the player to choose whether or not they flip a coin after each die roll where if they get heads they will advance one square and if they get tails they will go back one square. The strategy the player employs to decide when to flip the coin has a significant impact on the average duration of the game. We analyze six different non-trivial strategies.

Limit Cases And Strategy In Chutes and Ladders

Abstract

We analyze what happens to the average duration of a game of Chutes and Ladders as the probability of rolling approaches 100%. We utilize Markov models, and Monte Carlo simulations in Python. We also introduce strategy to the board game by allowing the player to choose whether or not they flip a coin after each die roll where if they get heads they will advance one square and if they get tails they will go back one square. The strategy the player employs to decide when to flip the coin has a significant impact on the average duration of the game. We analyze six different non-trivial strategies.
Paper Structure (14 sections, 38 equations, 16 figures)

This paper contains 14 sections, 38 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Modelling The Game As A Markov Chain
  3. Calculating the Probability the Game Will Last $M$ Rounds
  4. Calculating The Expected Duration of The Game $\sigma$
  5. Fully Weighted Die Cases
  6. Fully Weighted on 1,2,3, or 6
  7. Square Classifications
  8. Fully Weighted on 5
  9. Fully Weighted on 4
  10. Open Question
  11. Introducing Strategy
  12. Strategy Descriptions
  13. Determining $\sigma$ of Each Strategy
  14. SageMath & Python Code

Figures (16)

  • Figure 1: Chutes & Ladders Board
  • Figure 2: Markov Model at State 0, Version I
  • Figure 3: Markov Model at State 0, Version II
  • Figure 4: Markov Model of First Two Turns, Rolling From 14 and 38 Excluded
  • Figure 5: $P(M)$ For $1 \leq M \leq 100$
  • ...and 11 more figures