The 8 Puzzle Problem in AI

How AI Techniques Are Used to Solve the 8 Puzzle Problem

The 8 Puzzle Problem is solved using various AI techniques that explore the possible moves to reach the goal state. The primary AI methods used includes search algorithms and heuristic functions.

1. State Space Search

The 8 Puzzle Problem is a problem where each possible configuration of the puzzle is considered a "state." The search algorithms then explore these states to find a path from the initial state to the goal state.

2. Search Algorithms

Several search algorithms can be applied to solve the 8 Puzzle Problem:

• Breadth-First Search (BFS): This algorithm explores all possible moves at each depth level before moving on to the next level. It is guaranteed to find the shortest solution, but it can be slow and memory-intensive.
• Depth-First Search (DFS): This algorithm explores as far as possible along one branch before backtracking. It uses less memory than BFS but may not be able to find the shortest path.
• A Search:* This is one of the most efficient search algorithms used for the 8 Puzzle Problem. It combines of both BFS and DFS by using a heuristic function to guide the search towards the goal.

3. Heuristic Functions

Heuristics are used in AI to estimate how close a current state is to the goal state. The two most common heuristic functions used in the 8 Puzzle Problem are:

• Manhattan Distance: This function calculates the sum of the vertical and horizontal distances of each tile from its end position. It effectively guides the search by prioritizing states that are closer to the goal.

Manhattan Distance = |currentX - goalX| + |currentY - goalY|

You can also try this code with Online Java Compiler

Run Code

• Misplaced Tiles: This function counts the number of tiles that are not in their correct position compared to the goal state. While it is more simpler than Manhattan distance, but it may not be able to provide the most efficient path to the solution always.

4. Cost Functions

In algorithms like A*, a cost function combines the depth of the current state (i.e., how far it is from the initial state) and the heuristic value to prioritize which states to explore next. This approach helps in finding the shortest possible solution efficiently.

Heuristics for the 8 Puzzle Problem

Heuristics help in solving the 8-puzzle problem efficiently by estimating how far the current state is from the goal state. Two commonly used heuristics are:

1. Misplaced Tile Heuristic (h₁)

Counts the number of tiles that are not in their correct position.

Simple but less accurate.

Example:

If the tile 8 and the blank _ are misplaced, h₁ = 1.

Start State:    Goal State:
1  2  3        1  2  3
4  5  6        4  5  6
7  8  _        7  8  _

2. Manhattan Distance Heuristic (h₂)

Calculates the sum of the vertical and horizontal distances each tile needs to move to reach its correct position.

More accurate than misplaced tile heuristic.

Example: If tile 8 is two moves away from its correct position, it contributes 2 to h₂.

Implementation

Using A* Search Algorithm

The A* Search Algorithm is one of the most efficient ways to solve the 8 Puzzle Problem. It uses a heuristic function to estimate the cost of reaching the goal from the current state.

A* Algorithm Implementation

Here is a simple Python implementation of the A* Search Algorithm for solving the 8 Puzzle Problem:

• Python

Python

import heapq

class PuzzleState:
    def __init__(self, board, parent, move, depth, cost):
        self.board = board
        self.parent = parent
        self.move = move
        self.depth = depth
        self.cost = cost

    def __lt__(self, other):
        return self.cost < other.cost

def manhattan_distance(board, goal):
    distance = 0
    for i in range(1, 9):
        x1, y1 = divmod(board.index(i), 3)
        x2, y2 = divmod(goal.index(i), 3)
        distance += abs(x1 - x2) + abs(y1 - y2)
    return distance

def a_star_search(initial_state, goal_state):
    explored = set()
    priority_queue = []
    initial = PuzzleState(initial_state, None, None, 0, manhattan_distance(initial_state, goal_state))
    heapq.heappush(priority_queue, initial)

    while priority_queue:
        current_state = heapq.heappop(priority_queue)
        explored.add(tuple(current_state.board))

        if current_state.board == goal_state:
            return current_state

        for move in possible_moves(current_state.board):
            new_board = apply_move(current_state.board, move)
            if tuple(new_board) not in explored:
                new_state = PuzzleState(new_board, current_state, move, current_state.depth + 1, current_state.depth + 1 + manhattan_distance(new_board, goal_state))
                heapq.heappush(priority_queue, new_state)

    return None

def possible_moves(board):
    moves = []
    empty_index = board.index(0)
    if empty_index % 3 > 0:  # Move left
        moves.append(-1)
    if empty_index % 3 < 2:  # Move right
        moves.append(1)
    if empty_index > 2:  # Move up
        moves.append(-3)
    if empty_index < 6:  # Move down
        moves.append(3)
    return moves

def apply_move(board, move):
    new_board = board[:]
    empty_index = new_board.index(0)
    new_index = empty_index + move
    new_board[empty_index], new_board[new_index] = new_board[new_index], new_board[empty_index]
    return new_board

# Example usage
initial = [1, 2, 3, 4, 5, 6, 0, 7, 8]
goal = [1, 2, 3, 4, 5, 6, 7, 8, 0]
result = a_star_search(initial, goal)
if result:
    print("Puzzle solved!")
else:
    print("No solution found.")

You can also try this code with Online Python Compiler

Run Code

Output

Puzzle solved!

Solvability of the 8 Puzzle Problem

The 8-puzzle problem is solvable if the given initial state can be transformed into the goal state by only using valid moves (up, down, left, right). The solvability is determined using inversions.

1. What is an Inversion?

An inversion occurs when a larger number appears before a smaller number in a single row representation of the puzzle (excluding the blank space _).

2. Solvability Rule for an 8-Puzzle

• Convert the 3×3 grid into a 1D array (ignoring the blank tile _).
• Count the number of inversions (pairs where a larger number appears before a smaller one).
• If the number of inversions is even, the puzzle is solvable; otherwise, it is unsolvable.

Example

Start State:    Goal State:
1  8  2        1  2  3
_  4  3        4  5  6
7  6  5        7  8  _

• 1D representation (ignoring _): [1, 8, 2, 4, 3, 7, 6, 5]
• Inversions: (8,2), (8,4), (8,3), (8,7), (8,6), (8,5), (4,3), (7,6), (7,5), (6,5) → Total = 10 (even)
• Since 10 is even, this puzzle is solvable.

Explanation of Code

• PuzzleState Class: This class represents the state of the puzzle, including the board configuration, the parent state, the move that led to this state, the depth, and the cost.
   
• Manhattan Distance: This function calculates the heuristic value for the A* algorithm, representing the distance between the current state and the goal state.
   
• A Search Function:* This function implements the A* search algorithm. It explores all possible states and uses the Manhattan distance to guide the search towards the goal.

Key Features of the 8 Puzzle Problem

1. State Space Representation: Each possible arrangement of the puzzle is a state in the state space.
    
2. Goal State: The final arrangement of the puzzle that solves the problem.
    
3. Heuristic Search: Using functions like Manhattan distance to guide the search efficiently.
    
4. Move Generation: Sliding tiles into the empty space generates new states.
    
5. Optimality: Algorithms like A* can guarantee an optimal solution if the heuristic is admissible.

Frequently Asked Questions

What is the 8-puzzle problem in AI theory?

The 8-puzzle problem is a sliding tile puzzle with a 3×3 grid, where the goal is to reach a specific arrangement using valid moves.

Is the 8-puzzle always solvable?

No, the 8-puzzle is solvable only if the number of inversions is even in its linear representation. Otherwise, it is unsolvable.

What is the 8 problem-solving method?

Common methods include *Breadth-First Search (BFS), Depth-First Search (DFS), A Algorithm (using heuristics like Manhattan Distance), and Iterative Deepening Search (IDS)**.

Conclusion

The 8 Puzzle Problem is an example of how AI techniques, such as search algorithms, can be applied to solve real-world problems. Understanding and implementing solutions like the A* Search Algorithm not only deepens your knowledge of AI but also improves your problem-solving skills. Whether you're a beginner or an advanced learner, tackling the 8 Puzzle Problem will give you a solid foundation in AI concepts.