Question

How many solutions are there for the 8-Queen problem on an $8 \times 8$ chessboard?

• A. 12
• B. 91
• C. 92
• D. 64

Hint:

For the N-Queen problem, the total number of solutions increases rapidly with N. The 8-Queen problem has 92 solutions in total, but only 12 if we exclude symmetric duplicates.

Answer 1

The Correct Option is C

The 8-Queen problem is a famous constraint satisfaction problem in computer science and mathematics. The goal is to place 8 queens on a standard \(8 \times 8\) chessboard so that no two queens attack each other. That is, no two queens share:

• the same row
• the same column
• the same diagonal

To solve this, we explore permutations of queen placements, where each queen must occupy a unique row and column. The backtracking algorithm is a common approach used for generating all valid configurations.

Key Concepts:

• There are \(8!\) (40,320) possible ways to place one queen in each row with one queen per column.
• Many of these arrangements are invalid due to diagonal conflicts.
• The backtracking algorithm checks each configuration and prunes it when a conflict is detected, thus reducing the search space significantly.

Backtracking Algorithm Steps:

1. Start in the first row and try placing a queen in each column one by one.
2. For each placement, move to the next row and repeat the process, skipping columns that are already attacked.
3. Use recursion to backtrack when no valid position exists in the current row.
4. Each time all 8 queens are placed without conflict, a valid solution is found and counted.

By executing this algorithm, it is found that there are exactly:

92 valid configurations

These include all symmetric (rotated and reflected) solutions. If symmetry is considered (i.e., removing duplicates due to rotation and reflection), only 12 fundamentally distinct solutions exist.

Final Answer: \( \boxed{92} \)