Number of Valid Move Combinations On Chessboard

Number of Valid Move Combinations On Chessboard

What is this problem about?

The Number of Valid Move Combinations On Chessboard problem places chess pieces (rooks, queens, bishops) on an 8×8 board and asks how many ways all pieces can be moved simultaneously such that no two pieces occupy the same cell at any point during movement. This hard coding problem uses backtracking over all possible (start → end, speed) combinations for each piece.

Why is this asked in interviews?

PayPal asks this because it requires modeling piece movement paths, detecting collisions along paths (two pieces at the same cell at the same time step), and backtracking over invalid combinations. The array, backtracking, string, and simulation interview pattern is fully exercised.

Algorithmic pattern used

Backtracking over movement assignments. For each piece, enumerate all possible end positions reachable via legal moves (rook moves in straight lines, bishop on diagonals, queen combines both). Assign a "move plan" (path + speed) to each piece. Check if any two pieces collide (same cell at same time step) by simulating their movements step by step. Use backtracking to count valid non-colliding combinations.

Example explanation

Two rooks at (1,1) and (2,2). Possible moves include staying, or moving along rows/columns. For each combination of rook1's end position and rook2's end position, simulate step-by-step to check if paths cross at the same step. Count valid combinations.

Common mistakes candidates make

• Not checking intermediate cells (only checking final positions misses path crossings).
• Not including "stay in place" as a valid move.
• Incorrect path generation for each piece type.
• Not handling the case where two pieces swap positions (they pass through each other at a midpoint).

Interview preparation tip

Hard simulation+backtracking problems require clean separation of: (1) move generation per piece, (2) collision detection between two piece paths, and (3) backtracking over all piece-pair combinations. Test collision detection independently before integrating into backtracking. For chessboard problems, always enumerate piece moves as (delta_row, delta_col) direction vectors times step counts.