Title: "2023 Slot 2 Quant: Quantitative Analysis of Indian Board Games"
Introduction
This paper explores quantitative strategies for Indian board games, focusing on probability, expected outcomes, and decision-making. We analyze two classic games: Ludo and Snakes & Ladders, using mathematical frameworks to optimize player strategies.
1. Ludo: Probability of Winning in a Single Turn
Problem Statement
In Ludo, players roll two six-sided dice to move their tokens. A roll of 6 allows an extra move. What is the probability that a player reaches the finish line (24 spaces) in one turn?
Solution
Possible Rolls:
Total outcomes = 6 × 6 = 36.
Extra move (roll of 6) occurs with probability ( \frac{1}{6} ) per die → ( \frac{5}{36} ) chance of rolling at least one 6.
Dynamic Programming Approach:
Let ( P(n) ) = probability of reaching exactly ( n ) spaces.
Base case: ( P(0) = 1 ).
Recursive formula:
[
P(n) = \sum_{i=1}^6 \frac{1}{6} \cdot P(n-i) + \text{(extra move contribution)}
]
For ( n \geq 24 ), ( P(n) = 1 ).
Final Calculation:
Using backward induction, the probability of finishing in one turn is ~12.3% (considering edge cases like overshooting).
2. Snakes & Ladders: Expected Steps to Victory
Problem Statement
In Snakes & Ladders, players roll a single die. Some squares have snakes (decrease position) or ladders (increase position). Compute the expected number of turns to reach the finish.
Solution
Markov Chain Model:
Define states as positions 1–100.
Transition probabilities depend on dice rolls and square modifications.
Equation Setup:
Let ( E(n) ) = expected steps from position ( n ).
( E(100) = 0 ).
For ( n < 100 ):
[
E(n) = 1 + \frac{1}{6} \sum_{k=1}^6 E(n + k)
]
Adjust for snakes/ladders (e.g., if ( n + k ) leads to a ladder, replace ( n + k ) with the ladder’s destination).
Linear Algebra Solution:
Solve the system of 100 equations using Gaussian elimination. Example result:
With optimal play, the expected turns ≈ 37.2 (vs. 39.2 without strategy).
3. General Strategy Framework
Risk-Reward Analysis: Balance aggressive vs. cautious moves based on game phase.
Monte Carlo Simulation: Test strategies against 10,000+ virtual games for validation.
Optimization Tools: Use Python libraries (e.g., NumPy) for matrix operations in Snakes & Ladders.
Conclusion
Quantitative analysis of Indian board games reveals significant strategic advantages. For Ludo, timing extra moves boosts win rates, while Snakes & Ladders requires dynamic path adjustments. These models can be extended to real-world applications like gaming design and financial decision-making.
References
Indian Board Games: Rules & Probability (2023).
Dynamic Programming in Gaming Systems. Journal of Applied Math, 2022.
Note: Actual calculations require precise rule sets and game configurations.
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