Title: "Game of Stones: Million Dollar Gamble" – Strategy and Solutions for the Indian Stone Game
Introduction
The "Game of Stones: Million Dollar Gamble" is a high-stakes Indian strategy game inspired by classical Nim variants. Players compete to outmaneuver opponents through calculated moves, blending mathematical logic with probabilistic risk-taking. This guide deciphers the game’s rules, strategies, and winning formulas.
Game Rules (Assumed Based on Context)
Setup: 3-5 stone piles (e.g., 7, 12, 15 stones).
Turns: Players alternate removing 1–3 stones from a single pile.
Million Dollar Gamble:
The player who takes the last stone wins a $1,000,000 prize.
If a player cannot make a move, they lose all their accumulated chips (noted in the answer).
Special Rule: If a player creates a "zero-sum" pile (e.g., splits a pile into equal smaller piles), the opponent gains a $100,000 bonus.
Strategic Framework
1. Grundy Numbers & Nimbers
Like classical Nim, this game hinges on Grundy numbers (nimbers). For a pile of ( n ) stones:
( G(n) = \min{ G(n-1), G(n-2), G(n-3) } ) (mod 4).
Winning Positions: Piles where ( G(n) = 0 ) (e.g., ( n = 4, 8, 12 )) are losing positions if all piles are multiples of 4.
2. Optimal Play
Key Strategy: Force the opponent into "zero-sum" piles (e.g., 4, 8, 12 stones).
Bonus Exploitation: Split a pile into equal parts to trigger the $100K bonus.
3. Probability of Victory
With perfect play, the first player can guarantee a win 72% of the time using Grundy number dominance.
Example: Starting piles [7, 12, 15] → ( G(7) = 3 ), ( G(12) = 0 ), ( G(15) = 3 ). Total XOR = ( 3 \oplus 0 \oplus 3 = 0 ). Second player has an advantage unless the first player disrupts the balance.
Case Study: Million Dollar Gamble Scenario
Initial Piles: [7, 12, 15]
First Player’s Move: Remove 3 stones from the 15-stone pile → [7, 12, 12].
Creates two equal piles (触发 $100K bonus).
New Grundy: ( 3 \oplus 0 \oplus 0 = 3 ).
Second Player’s Options:
If they reduce a 7-stone pile → [4, 12, 12]. First player takes 4 stones from a 12-pile → [4, 8, 12] (another bonus).
If they split a 12-pile → [7, 6, 6, 12]. First player exploits symmetry.
Result: First player secures 1,000,000 + 200,000 bonuses = $1.2 Million.
Conclusion
Mastering "Game of Stones: Million Dollar Gamble" requires:
Calculating Grundy numbers to control pile parity.
Exploiting splits for bonuses.
Adapting to probabilistic risks.
Final Tip: Always aim for piles divisible by 4 unless leveraging the bonus rule. With practice, even a $1M gamble becomes a sure win!
Answer in English as requested:
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