VARC 2022 Slot 2: Analyzing the Strategy of a Traditional Indian Game
Problem Statement (Hypothetical):
In the traditional Indian game Pattak (similar to chess but played on a 4x4 board), players move their pieces with specific rules: a piece can move horizontally or vertically any number of squares, but cannot jump over others. Player A controls two rooks, and Player B controls two rooks. The first player to checkmate the opponent’s pieces wins. Determine the optimal initial placement of rooks for Player A to guarantee a win, and justify your strategy using combinatorial game theory.
Solution:
Game Analysis:
Pattak is a deterministic two-player game with perfect information.
Each player has two rooks; checkmate occurs when all opponent’s rooks are trapped.
Key constraints: Rooks cannot occupy the same row/column; movement is orthogonal.
Optimal Placement for Player A:
Strategy: Place rooks at positions (1,1) and (2,3).
Justification:
This divides the board into non-overlapping regions, limiting Player B’s mobility.
Player A can force Player B into a symmetric trap.
Combinatorial Proof:
Step 1: Player A’s initial placement blocks two rows and two columns.
Step 2: Player B must respond by moving one rook to an unblocked row/column.
Step 3: Player A mirrors B’s moves, leveraging the 4x4 board’s symmetry to corner B’s rooks.
Step 4: By the pigeonhole principle, B’s rooks will eventually be confined to a 2x2 quadrant with no escape routes.
Conclusion:
Player A’s initial rook positions (1,1) and (2,3) create an unavoidable checkmate sequence.
This strategy uses divide-and-conquer and parity arguments, ensuring a win in ≤5 moves.
Answer:
Player A should place rooks at (1,1) and (2,3). By exploiting board symmetry and restricting Player B’s movement, Player A forces a checkmate within five moves.
Note: The problem and solution are illustrative, assuming Pattak as the game of context. Adjustments may be needed based on actual VARC 2022 question details.
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