Number of Valid Move Combinations On Chessboard

Number of Valid Move Combinations On Chessboard - Problem

There is an 8 x 8 chessboard containing n pieces (rooks, queens, or bishops). You are given a string array pieces of length n, where pieces[i] describes the type of the i^th piece. In addition, you are given a 2D integer array positions also of length n, where positions[i] = [ri, ci] indicates that the i^th piece is currently at the 1-based coordinate (ri, ci) on the chessboard.

When making a move for a piece, you choose a destination square that the piece will travel toward and stop on.

A rook can only travel horizontally or vertically from (r, c) to the direction of (r+1, c), (r-1, c), (r, c+1), or (r, c-1).
A queen can only travel horizontally, vertically, or diagonally from (r, c) to the direction of (r+1, c), (r-1, c), (r, c+1), (r, c-1), (r+1, c+1), (r+1, c-1), (r-1, c+1), (r-1, c-1).
A bishop can only travel diagonally from (r, c) to the direction of (r+1, c+1), (r+1, c-1), (r-1, c+1), (r-1, c-1).

You must make a move for every piece on the board simultaneously. A move combination consists of all the moves performed on all the given pieces. Every second, each piece will instantaneously travel one square towards their destination if they are not already at it. All pieces start traveling at the 0th second. A move combination is invalid if, at a given time, two or more pieces occupy the same square.

Return the number of valid move combinations.

Note: No two pieces will start in the same square. You may choose the square a piece is already on as its destination. If two pieces are directly adjacent to each other, it is valid for them to move past each other and swap positions in one second.

Input & Output

Example 1 — Basic Two Pieces

$ Input: pieces = ["rook","bishop"], positions = [[1,1],[1,8]]

› Output: 15

💡 Note: Rook at (1,1) can move horizontally/vertically, bishop at (1,8) can move diagonally. They have multiple valid combinations without collision.

Example 2 — Single Piece

$ Input: pieces = ["queen"], positions = [[1,1]]

› Output: 22

💡 Note: A single queen at (1,1) can stay in place or move to any of 21 valid squares on the 8x8 board.

Example 3 — Adjacent Pieces

$ Input: pieces = ["rook","rook"], positions = [[1,1],[1,2]]

› Output: 49

💡 Note: Two rooks adjacent horizontally can move independently in many directions without collision.

Constraints

1 ≤ pieces.length ≤ 4
pieces[i] is either "rook", "queen", or "bishop"
There will be at most one piece of each type on the board
1 ≤ x_i, y_i ≤ 8
All pieces are on distinct squares

Visualization

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Understanding the Visualization

Input

Chess pieces with their starting positions

Process

Generate all valid move combinations with collision checking

Output

Count of valid move combinations

Key Takeaway

🎯 Key Insight: Simulate piece movement step-by-step in time to detect collisions during transit, not just at final positions

Asked in

G Google 12 f Facebook 8 M Microsoft 6

The key insight is to use backtracking with simulation to check all possible move combinations while pruning invalid paths early. Best approach is optimized backtracking that generates moves piece by piece and detects collisions during path simulation. Time: O(8^n), Space: O(n × max_distance)

Common Approaches

Approach	Time	Space	Notes
✓ Optimized Backtracking with Pruning	O(8^n) with pruning	O(n × max_distance)	Use backtracking to generate moves with early collision detection
Brute Force with Full Simulation	O(8^n × max_distance)	O(n × max_distance)	Generate all possible move combinations and simulate each one completely

Optimized Backtracking with Pruning — Algorithm Steps

Generate moves for one piece at a time
Check for early collision detection
Prune invalid branches immediately
Continue recursion only for valid paths

Visualization

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Step-by-Step Walkthrough

Generate Move

Try one move for current piece

Check Collision

Test if path conflicts with existing pieces

Prune/Continue

Stop if collision found, otherwise recurse

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

int solution(char pieces[][10], int positions[][2], int n) {
    // Simplified implementation
    return 1;
}

int main() {
    char pieces[10][10];
    int positions[10][2];
    int n = 1;
    
    strcpy(pieces[0], "rook");
    positions[0][0] = 1;
    positions[0][1] = 1;
    
    printf("%d\n", solution(pieces, positions, n));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(8^n) with pruning

Backtracking with early pruning reduces the search space significantly

✓ Linear Growth

Space Complexity

O(n × max_distance)

Recursion stack depth and move storage for n pieces

⚡ Linearithmic Space

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Medium Frequency

~35 min Avg. Time

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Smart Actions

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Optimized Backtracking with Pruning — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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