Maximum Number of Moves to Kill All Pawns

Maximum Number of Moves to Kill All Pawns

What is this problem about?

The Maximum Number of Moves to Kill All Pawns coding problem places a knight on a chessboard at a given position, with pawns scattered across the board. Two players alternate turns. On each turn, the current player moves the knight to the nearest pawn (by BFS moves) and removes it. The first player maximizes total moves; the second player minimizes. Find the total number of knight moves made, assuming both play optimally.

Why is this asked in interviews?

Google uses this problem because it combines BFS for shortest knight paths, bitmask DP for subset states, and game theory (minimax). It's a complete challenge testing multiple algorithmic competencies simultaneously. Candidates must precompute pawn-to-pawn distances and then apply game-theory DP on the bitmask of remaining pawns.

Algorithmic pattern used

BFS precomputation + Bitmask Game Theory DP:

1. BFS from the starting position and each pawn to compute shortest knight-move distances to all other positions.
2. Define dp[mask][last] = optimal total moves when the set of remaining pawns is mask and the knight is currently at pawn last. Player 1 maximizes, player 2 minimizes, alternating based on the number of pawns already captured (parity of bits cleared).
3. Transition: for each remaining pawn p in mask, dp[mask][last] = best(dp[mask ^ (1<<p)][p] + dist[last][p]) where best = max if current turn is player 1, min if player 2.

Example explanation

Starting pos: (0,0). Pawns: [(1,2), (2,4)].

• BFS gives distances: start→pawn0=1 move, start→pawn1=4 moves, pawn0→pawn1=3 moves.
• Player 1 maximizes: go to pawn1 first (4 moves), then player 2 minimizes from pawn1→pawn0 (3 moves). Total = 7.
• Or: Player 1 goes to pawn0 (1 move), player 2 goes to pawn1 (3 moves). Total = 4.
• Player 1 picks max: 7.

Common mistakes candidates make

• Not precomputing all pairwise distances: Recomputing BFS during DP is too slow.
• Wrong turn assignment: Turn depends on the number of pawns captured so far (parity). Getting this wrong flips max/min decisions.
• BFS on wrong movement: Knight moves in L-shapes — 8 possible directions. Using grid adjacency instead gives wrong distances.

Interview preparation tip

For the Array Math Bitmask Breadth-First Search Game Theory Bit Manipulation interview pattern, this problem combines three independent skills: BFS, bitmask DP, minimax. Master each separately before combining. The bitmask DP for TSP-like problems (visiting all nodes) with game theory is a common hard interview pattern at Google.