15 puzzle solver

The 15 puzzle (also called the Sliding Puzzle) is a classic sliding tile game with 15 square tiles numbered 115 arranged in a 44 grid, with one empty space. The goal is to rearrange the tiles from a scrambled initial state into numerical order (left to right, top to bottom) by sliding them into the empty space. How to Solve the 15 Puzzle There are multiple approaches to solving the 15 puzzle, ranging from simple human strategies to advanced algorithms. Human-Friendly Solving Methods These strategies work well for manually solving the puzzle: A. Layer-by-Layer (Beginner's Method) Solve the first row (1, 2, 3, 4) - Move tiles one by one into their correct positions. - Use the empty space to maneuver tiles. Solve the second row (5, 6, 7, 8) - Avoid disturbing the first row while moving tiles. Solve the third and fourth rows (915) - Use a combination of sliding and "looping" moves to position the remaining tiles. B. Corner-Building (Intermediate Method) Fix the first three tiles of the top row (1, 2, 3). Place the 4th tile below its correct position and rotate it into place. Repeat for the next rows, keeping solved tiles intact. Algorithmic Approaches (For Programmatic Solvers) For writing a solver program, you can use: A. Breadth-First Search (BFS) - Guarantees the shortest solution but is slow for complex puzzles. - Uses a queue to explore all possible moves level by level. B. A Search (Optimal for 15 Puzzle) - Uses a heuristic (like Manhattan Distance or Misplaced Tiles) to prioritize moves. - More efficient than BFS for larger puzzles. C. IDA (Iterative Deepening A) - A memory-efficient variant of A that works well for the 15 puzzle. Example Python Solver (Using A with Manhattan Distance) Key Notes: Not all initial states are solvable. A puzzle is solvable if: - The number of inversions (tiles out of order) is even for the empty space in the last row. - If the empty space is in the 3rd row, the number of inversions must be odd. Optimal solvers (A, IDA) guarantee the shortest solution. For manual solving, practice helps!* The more you solve, the better you recognize patterns. Would you like help with a specific puzzle configuration?