Minimum Moves to Capture The Queen

The goal is to capture the black queen in the minimum number of moves using only the white pieces. Typically, you can achieve this in one or two moves depending on initial positions. By analyzing all reachable squares for the rook and bishop, you can enumerate possible attacks and identify the optimal sequence efficiently.

Problem Statement

You are given a standard 8x8 chessboard with three pieces: a white rook, a white bishop, and a black queen. Each piece's position is given as integers a, b for the rook, c, d for the bishop, and e, f for the black queen. No two pieces occupy the same square, and the board is 1-indexed from top-left to bottom-right.

Your task is to determine the minimum number of moves required to capture the black queen, moving only the white pieces. Each move consists of a legal chess move for the rook or bishop, and you must consider all possible positions to find the optimal sequence that guarantees capturing the queen.

Examples

Example 1

Input: a = 1, b = 1, c = 8, d = 8, e = 2, f = 3

Output: 2

We can capture the black queen in two moves by moving the white rook to (1, 3) then to (2, 3). It is impossible to capture the black queen in less than two moves since it is not being attacked by any of the pieces at the beginning.

Example 2

Input: a = 5, b = 3, c = 3, d = 4, e = 5, f = 2

Output: 1

We can capture the black queen in a single move by doing one of the following:

• Move the white rook to (5, 2).
• Move the white bishop to (5, 2).

Constraints

• 1 <= a, b, c, d, e, f <= 8
• No two pieces are on the same square.

Solution Approach

Direct Attack Check

First, check if either the rook or bishop can capture the queen in a single move. For the rook, verify row and column alignment without obstruction. For the bishop, check diagonal alignment. If one piece can capture immediately, the answer is 1.

One-Step Reach Enumeration

If a direct capture is impossible, enumerate all reachable positions for each white piece in one move. For each reachable square, check if moving there allows the next move to capture the queen. This relies on simple math calculations for row, column, and diagonal alignment.

Combine Moves and Minimize

Simulate sequences where one piece moves first and then the second piece moves to capture the queen. Track the minimum number of moves across all sequences. The pattern here is math plus enumeration: calculate reachable squares and enumerate attack possibilities to ensure the optimal two-move solution is found.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on enumerating all reachable squares for both white pieces and evaluating attack paths, which is bounded by the fixed 8x8 board. Space complexity is minimal, storing positions and sequences for analysis.

What Interviewers Usually Probe

• Can you determine if a single move capture is possible?
• How do you enumerate all reachable squares for the rook and bishop efficiently?
• What is the strategy to minimize moves when direct capture is blocked?

Common Pitfalls or Variants

Common pitfalls

• Forgetting the board is 1-indexed, causing off-by-one errors in position checks.
• Overlooking diagonal paths for the bishop when enumerating potential moves.
• Assuming capture always takes two moves without checking immediate alignment opportunities.

Follow-up variants

• Change board size to NxN and analyze minimum moves using the same enumeration strategy.
• Include additional white pieces and determine the fewest moves using multiple piece types.
• Require capturing multiple black pieces, extending the enumeration pattern to sequences of attacks.

FAQ

What is the main strategy to solve Minimum Moves to Capture The Queen?

Use math to check alignment and enumerate reachable squares for the rook and bishop to determine one- or two-move captures.

Can the queen be captured in one move every time?

No, capture depends on initial positions; only if the rook or bishop is aligned without obstruction.

Does the solution require simulating all board positions?

Only reachable positions for the white pieces need simulation, not every square on the board.

Why use enumeration for this problem?

Enumeration ensures all attack sequences are considered, identifying the minimal number of moves reliably.

How does math plus enumeration pattern apply here?

Math checks alignment and movement rules, while enumeration explores all possible sequences to guarantee minimum moves.