The N-Queens problem is a classic example of a combinatorial puzzle that has been widely studied in the fields of computer science, mathematics, and artificial intelligence. It's a problem that illustrates the concept of constraint satisfaction and is used to showcase algorithms that can handle such constraints efficiently. To understand the N-Queens problem, let's break it down from its first principles.

What is the N-Queens Problem?

The N-Queens problem involves placing N chess queens on an N×N chessboard so that no two queens threaten each other. This means that no two queens can be placed in the same row, column, or diagonal. The challenge stems from the queen's ability in chess to move any number of squares vertically, horizontally, or diagonally.

Fundamental Concepts

  1. Chessboard: A square grid where the game of chess is played. In the context of the N-Queens problem, its size is N×N, where N is the number of queens to be placed.

  2. Queen: A chess piece that can move any number of squares along a row, column, or diagonal.

  3. Constraint Satisfaction: This is a key concept where the solution to a problem must satisfy a number of constraints or limitations. In the N-Queens problem, the constraints are that no two queens can attack each other.

Solving the N-Queens Problem

To solve the N-Queens problem, one must find a way to place N queens on the board without violating the attacking constraint. The complexity of the problem increases exponentially with each additional queen, making it a non-trivial problem for large N.

Methods of Solution

Several approaches can be used to solve the N-Queens problem:

  1. Brute Force: Trying every possible arrangement of queens on the board. This method is impractical for large N due to its exponential time complexity.

  2. Backtracking: A more efficient method that places queens one by one in different rows and backtracks whenever a position leads to a violation of the constraints.

  3. Heuristic Algorithms: These include greedy algorithms, genetic algorithms, and others that use specific strategies to find a solution more efficiently than brute force.

  4. Constraint Programming: This involves defining the N-Queens problem in terms of constraints and using constraint-solving techniques to find a solution.


For N=4, one possible solution on a 4x4 board is:

  • Q represents a queen, and * represents an empty space.
* Q * *
* * * Q
Q * * *
* * Q *

In this arrangement, no two queens can attack each other.


The N-Queens problem is more than just a puzzle; it serves as a benchmark for testing algorithms that solve constraint satisfaction problems. Its solutions and the methods developed to solve it have applications in scheduling, planning, and the design of computer systems and networks.

Understanding the N-Queens problem from first principles helps in grasping the complexity of constraint satisfaction problems and the effectiveness of various algorithms in solving them. It's a fundamental problem that bridges the gap between theoretical computer science and practical algorithm design and analysis.