N-Queens Problem Visualizer

Visualize the N-Queens problem solution using backtracking algorithm. Place N queens on an N×N chessboard so that no two queens threaten each other.

Configuration

The N-Queens Problem:
Place N chess queens on an N×N board so that no two queens attack each other

About N-Queens

The N-Queens problem is a classic puzzle where you must place N chess queens on an N×N chessboard.

Rules:

• No two queens can be in the same row

• No two queens can be in the same column

• No two queens can be on the same diagonal

Complexity:

This is an NP-hard problem solved using backtracking

Features

  • Interactive board visualization
  • Find all solutions
  • Animate through solutions
  • Adjustable animation speed
  • Show attacked cells
  • Copy solution positions
  • Download all solutions
  • Support 1×1 to 12×12 boards

Solution Counts

4×4:2 solutions
5×5:10 solutions
6×6:4 solutions
8×8:92 solutions
10×10:724 solutions

Algorithm

Backtracking:

Places queens row by row

Pruning:

Skips invalid positions early

Recursion:

Explores all valid placements

Time Complexity:

O(N!) worst case

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About N-Queens Problem Visualizer

Our N-Queens Problem Visualizer helps you understand and solve one of the most famous problems in computer science. Watch as the backtracking algorithm finds all possible ways to place N chess queens on an N×N chessboard so that no two queens attack each other. Perfect for students learning algorithms, developers preparing for interviews, and educators teaching problem-solving techniques.

How to Use the N-Queens Visualizer

  1. Enter board size (1 to 12) or click a quick example button
  2. Set animation speed in milliseconds (100-5000ms)
  3. Click "Solve" to find all solutions
  4. Navigate through solutions using Previous/Next buttons
  5. Click "Animate" to automatically cycle through solutions
  6. View queen positions and attacked cells
  7. Copy individual solution or download all solutions
  8. Click "Reset" to start over with a new configuration

What is the N-Queens Problem?

The N-Queens problem is a classic puzzle where you must place N chess queens on an N×N chessboard such that no two queens threaten each other. In chess, a queen can attack any piece on the same row, column, or diagonal. The challenge is to find all possible arrangements where all queens are safe from each other.

The Three Rules

  • No Row Conflicts: No two queens can be in the same row
  • No Column Conflicts: No two queens can be in the same column
  • No Diagonal Conflicts: No two queens can be on the same diagonal (both directions)

Backtracking Algorithm

The N-Queens problem is solved using a backtracking algorithm, which is a systematic way to explore all possible solutions:

  1. Place Queens Row by Row: Start from the first row and try to place a queen
  2. Check Safety: For each column, check if placing a queen is safe
  3. Recurse: If safe, place the queen and move to the next row
  4. Backtrack: If no safe position exists, go back to the previous row
  5. Try Next Position: Try the next column in the previous row
  6. Record Solution: When all N queens are placed, save the solution
  7. Continue: Backtrack to find all other solutions

Solution Counts by Board Size

  • 1×1: 1 solution (trivial case)
  • 2×2: 0 solutions (impossible)
  • 3×3: 0 solutions (impossible)
  • 4×4: 2 solutions
  • 5×5: 10 solutions
  • 6×6: 4 solutions
  • 7×7: 40 solutions
  • 8×8: 92 solutions (classic chess board)
  • 9×9: 352 solutions
  • 10×10: 724 solutions
  • 11×11: 2,680 solutions
  • 12×12: 14,200 solutions

Complexity Analysis

  • Time Complexity: O(N!) in the worst case - factorial time
  • Space Complexity: O(N) for the recursion stack
  • Problem Class: NP-hard (no known polynomial-time solution)
  • Pruning: Backtracking significantly reduces the search space

Features

  • Interactive Visualization: See the chessboard and queen placements
  • All Solutions: Find every possible solution for any board size
  • Animation: Auto-cycle through solutions at adjustable speed
  • Visual Feedback: Highlighted attacked cells and safe positions
  • Position Display: View exact coordinates of each queen
  • Export Options: Copy or download solutions
  • Quick Examples: One-click loading of common board sizes
  • Responsive Design: Works on all devices

Historical Background

  • 1848: First published by chess player Max Bezzel as the "Eight Queens" puzzle
  • 1850: Franz Nauck extended it to the general N-Queens problem
  • 1874: First solutions published in a German chess magazine
  • Classic Problem: One of the most studied problems in computer science
  • Modern Research: Still actively studied for optimization techniques

Applications in Computer Science

  • Algorithm Education: Teaching backtracking and recursion concepts
  • Constraint Satisfaction: Example of CSP (Constraint Satisfaction Problem)
  • Search Algorithms: Demonstrating depth-first search with backtracking
  • Optimization: Studying pruning and search space reduction
  • Parallel Computing: Testing parallel search algorithms
  • AI Research: Benchmark for constraint solving techniques

Real-World Applications

  • Resource Allocation: Scheduling tasks without conflicts
  • VLSI Design: Placing components on circuit boards
  • Network Design: Avoiding interference in wireless networks
  • Timetabling: Creating schedules without conflicts
  • Graph Coloring: Related to vertex coloring problems
  • Puzzle Games: Basis for many logic puzzles

Optimization Techniques

  • Symmetry Reduction: Eliminate symmetric solutions to reduce computation
  • Bit Manipulation: Use bitwise operations for faster conflict checking
  • Heuristics: Try most constrained positions first
  • Parallel Search: Distribute search across multiple threads
  • Memoization: Cache results of subproblems

Interesting Facts

  • For an 8×8 board, there are 92 solutions (12 fundamental solutions when accounting for symmetry)
  • The largest board with all solutions counted is 27×27 with 234,907,967,154,122,528 solutions
  • No solution exists for 2×2 and 3×3 boards (mathematically impossible)
  • The problem becomes exponentially harder as N increases
  • Modern computers can solve up to N=27 in reasonable time

Learning Tips

  • Start with small boards (4×4 or 6×6) to understand the pattern
  • Watch the animation to see how backtracking works
  • Try to predict where the next queen will be placed
  • Notice how the algorithm backtracks when it hits a dead end
  • Compare solution counts for different board sizes
  • Implement your own version in your favorite programming language

Frequently Asked Questions

Why are there no solutions for 2×2 and 3×3 boards?

On a 2×2 board, any two queens will attack each other. On a 3×3 board, it's impossible to place 3 queens without at least two attacking each other due to the limited space and diagonal attacks.

How long does it take to solve larger boards?

Time increases exponentially. Boards up to 12×12 solve almost instantly. Beyond that, computation time grows rapidly. The 27×27 board took significant computing resources to solve completely.

Can this algorithm be improved?

Yes! Optimizations include using bitwise operations for faster conflict checking, symmetry reduction to eliminate duplicate solutions, and parallel processing to distribute the search.

Is this problem related to other puzzles?

Yes! The N-Queens problem is related to Sudoku, graph coloring, Latin squares, and many other constraint satisfaction problems. The backtracking technique applies to all of them.

Interview Preparation

The N-Queens problem is a popular coding interview question because it tests:

  • Understanding of backtracking algorithms
  • Recursion and base cases
  • Constraint checking logic
  • Problem-solving approach
  • Code organization and clarity
  • Optimization thinking