How to Solve the 8-Puzzle with Heuristic Search

TL;DR

Solve the 8-puzzle by generating legal successor states, counting each state's misplaced tiles against the goal, and exploring the state with the smallest heuristic value. This informed approach can reach the goal quickly while retaining alternatives in memory, but the heuristic does not guarantee that the resulting solution is optimal.

Transcript

Hello friends, welcome to Gate Smashers In this video we are going to discuss How to solve 8-Puzzle Problem with heuristic That is also called the informed search technique In last video we discussed That how do we solve a 8 puzzle problem without heuristic Which we call as blind search Or brute force method or you can call it with an uninformed wa... Read More

Key Insights

• A heuristic value is an estimate used to decide which puzzle state should be explored next. In the demonstrated best-first method, smaller values are preferred because they indicate that a state appears closer to the required goal arrangement.
• The misplaced-tiles heuristic is calculated by comparing every numbered tile in the current state with its corresponding position in the goal state. Each incorrectly positioned tile contributes to the value, while the empty space does not need to be counted.
• The empty space determines the legal successors of an 8-puzzle state. Depending on its location, it may move up, down, left, or right, and each legal move produces a new state that can be evaluated against the goal.
• The initial state has a heuristic value of 3 because tiles 4, 5, and 8 are misplaced relative to the goal. This value establishes an estimate of the starting state's distance from the desired arrangement.
• The first expansion creates three successor states with heuristic values of 4, 4, and 2. Best-first selection chooses the state valued at 2, while the two states valued at 4 are retained in memory instead of being expanded immediately.
• The next selected state can generate moves in four directions because its empty space is in the middle. One move recreates its parent and provides no benefit, while the remaining successors are evaluated using the same misplaced-tiles calculation.
• The most promising second-level successor has a heuristic value of 1. Expanding that state leads to the goal arrangement at depth 3, at which point the search stops because further exploration is unnecessary.
• Heuristic search can reduce unnecessary exploration and find a solution quickly, but it does not guarantee an optimal solution. The method may return an optimal result in a particular puzzle, yet its general promise is a good solution rather than assured optimality.

Questions & Answers

Q: How do you solve the 8-puzzle using heuristic search?

Start by comparing the current arrangement with the goal and counting the misplaced numbered tiles. Generate every state allowed by moving the empty space up, down, left, or right. Calculate the same heuristic for each successor, then expand the state with the smallest value. Repeat this process until a generated arrangement equals the goal state, then stop the search.

Q: What is the misplaced-tiles heuristic in the 8-puzzle?

The misplaced-tiles heuristic is the number of numbered tiles that are not in their required goal positions. Each current state is compared directly with the goal arrangement, and every incorrectly placed numbered tile is counted. The empty space does not need to be included. A smaller result is treated as a better estimate because the state appears closer to the goal.

Q: Why is the empty space excluded from the heuristic value?

The empty space can be excluded because the demonstrated heuristic focuses on numbered tiles that are misplaced relative to the goal. The lecture notes that the space could be counted, but doing so is unnecessary because it would be included consistently. Excluding it keeps the calculation focused on tiles 1 through 8 and does not change how the demonstrated successors are ranked.

Q: How are successor states generated in an 8-puzzle?

Successor states are generated by moving the empty space in every legal direction allowed by its current position. Possible directions are up, down, left, and right, although edge positions prevent some moves. Each valid movement swaps the empty space with an adjacent numbered tile, producing a new arrangement whose heuristic value can then be calculated against the goal.

Q: How does best-first search choose the next puzzle state?

Best-first search calculates a heuristic value for every generated successor and selects the state with the minimum value for expansion. In the example, the first three successors receive values of 4, 4, and 2, so the state valued at 2 is chosen. The alternatives are kept in memory, but they are not explored at that moment.

Q: What happens when a move recreates the parent state?

A move that recreates the parent state provides no progress because it returns the puzzle to an arrangement that has already been reached. The lecture shows that such a state may still be written among the possible moves, but it offers no benefit. The search instead evaluates and prioritizes successors that produce different arrangements and potentially smaller heuristic values.

Q: How quickly does the demonstrated heuristic path reach the goal?

The demonstrated path reaches the goal at depth 3. The search first chooses a successor with heuristic value 2, then selects a resulting state with heuristic value 1, and finally moves the empty space to produce the goal arrangement. Once the current state matches the goal, the process stops and does not explore any remaining alternatives.

Q: Does heuristic search guarantee an optimal 8-puzzle solution?

No, the heuristic method described does not guarantee that every solution will be optimal. It uses estimated closeness to guide the search and can therefore produce an answer quickly, sometimes even an optimal one. However, its general benefit is finding a good solution with less immediate exploration, not proving that the chosen path always has the lowest possible cost.

Summary & Key Takeaways

• Heuristic search solves the 8-puzzle by evaluating how close each state appears to the goal. The demonstrated heuristic counts misplaced numbered tiles while excluding the empty space. After generating every legal successor, the method selects the state with the smallest heuristic value for the next expansion.

• From the starting state, moving the empty space up, down, or right creates three successors with heuristic values of 4, 4, and 2. The state valued at 2 is expanded because it appears closest to the goal, while the other generated states remain available in memory.

• The selected path continues through a state with heuristic value 1 and reaches the goal at depth 3. Compared with blind search, this process avoids immediately expanding every state at each depth. It finds an answer quickly, although a heuristic method may produce a good solution without guaranteeing optimality.