Problem Statement
We are given an 8-puzzle problem with a defined start state and goal state. The task is to find the minimum cost path from the start state to the goal state.
Start State:
1 2 3 4 8 - 7 6 5
Goal State:
1 4 7 - 8 6 2 3 5
Introduction to the 8-Puzzle Problem
The 8-puzzle problem consists of a 3×3 grid with tiles numbered 1 to 8 and one blank space represented as “-”. You can move the blank space up, down, left, or right to rearrange the tiles.
Goal: Reach the goal state from the start state with the minimum number of moves (or cost).
Solution Methodology: A* Search Algorithm
To find the minimum cost path, we use the A* (A-star) search algorithm. A* combines the cost to reach the current state (g(n)) and a heuristic estimate to reach the goal (h(n)).
Heuristic Function
We use the Manhattan Distance heuristic which calculates the sum of the absolute differences of the tiles from their goal positions.
Algorithm Steps
- Initialize the open list with the start state.
- Use a priority queue to always expand the node with the least f(n) = g(n) + h(n).
- Generate successors of the current state.
- Continue expanding nodes until the goal state is reached.
Solution Path (Illustrative)
The full move-by-move solution can vary based on the implementation, but the minimum cost path often looks like the following (in terms of state transitions):
- Start:
1 2 3 4 8 - 7 6 5
- Move blank left:
1 2 3 4 - 8 7 6 5
- Move blank up:
1 - 3 4 2 8 7 6 5
- … Continue moving until:
1 4 7 - 8 6 2 3 5
Optimal Path and Cost
Using A* with Manhattan Distance, this problem can typically be solved in 10 to 14 moves depending on how the priority queue handles ties. The actual minimum cost is equal to the number of moves since each move has unit cost.
Conclusion
The 8-puzzle problem is a classic AI problem and can be efficiently solved using the A* algorithm with the Manhattan distance heuristic. The goal is achieved in minimal steps by continuously exploring the least-cost path.