Knight's Tour Problem
Find a sequence of moves for a knight to visit every square on a chessboard exactly once. Classic backtracking problem demonstrating exhaustive search with pruning.
Visualization
Interactive visualization for Knight's Tour Problem
Interactive visualization with step-by-step execution
Implementation
1function knightTour(n: number): number[][] | null {
2 const board: number[][] = Array(n).fill(0).map(() => Array(n).fill(-1));
3 const moveX = [2, 1, -1, -2, -2, -1, 1, 2];
4 const moveY = [1, 2, 2, 1, -1, -2, -2, -1];
5
6 board[0][0] = 0;
7
8 function isSafe(x: number, y: number): boolean {
9 return x >= 0 && x < n && y >= 0 && y < n && board[x][y] === -1;
10 }
11
12 function solveUtil(x: number, y: number, moveCount: number): boolean {
13 if (moveCount === n * n) return true;
14
15 for (let i = 0; i < 8; i++) {
16 const nextX = x + moveX[i];
17 const nextY = y + moveY[i];
18
19 if (isSafe(nextX, nextY)) {
20 board[nextX][nextY] = moveCount;
21
22 if (solveUtil(nextX, nextY, moveCount + 1)) return true;
23
24 board[nextX][nextY] = -1;
25 }
26 }
27
28 return false;
29 }
30
31 return solveUtil(0, 0, 1) ? board : null;
32}Deep Dive
Theoretical Foundation
Knight moves in L-shape: 2 squares in one direction, 1 in perpendicular. Must visit all n² squares exactly once. Backtrack when no valid moves available. Can use Warnsdorff's heuristic for better performance.
Complexity
Time
O(8^(n²))
O(8^(n²))
O(8^(n²))
Space
O(n²)
Applications
Industry Use
Chess puzzle generation and solving
Game development (movement pattern algorithms)
Robotics path planning with movement constraints
Educational tools for teaching backtracking
Puzzle and brain training applications
Algorithm visualization and demonstration
Constraint satisfaction problem examples
Use Cases
Related Algorithms
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The N-Queens Problem asks: place N chess queens on an N×N chessboard so that no two queens threaten each other. This means no two queens can be on the same row, column, or diagonal. It's a classic constraint satisfaction problem that demonstrates backtracking, pruning, and systematic exploration of solution spaces. The problem has applications in parallel processing, resource allocation, and algorithm design education.
Sudoku Solver
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Combination Sum
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