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8 Puzzle problem in Python

The 8 puzzle problem solution is covered in this article. A 3 by 3 board with 8 tiles (each tile has a number from 1 to 8) and a single empty space is provided. The goal is to use the vacant space to arrange the numbers on the tiles such that they match the final arrangement. Four neighbouring (left, right, above, and below) tiles can be moved into the available area.

For instance,

8 Puzzle problem in Python

1. DFS (Brute - Force) :

On the state-space tree (Set of all configurations of a particular issue, i.e., all states that may be reached from the beginning state), we can do a depth-first search.

8 Puzzle problem in Python

Figure : 8 Puzzle's State Space Tree

In this solution, further movements might not always send us closer to the objective, but rather further away. Regardless of the initial state, the state-space tree searches down the leftmost route from the root. With this method, an answer node might never be discovered.

2. BFS (Brute - Force) :

We can search the state space tree using a breadth-first approach. It always locates the goal state that is closest to the root. However, the algorithm tries the same series of movements as DFS regardless of the initial state.

3. Branch and Bound :

By avoiding searching in sub-trees which do not include an answer node, an "intelligent" ranking function, also known as an approximatsion costs function, may frequently speed up the search for an answer node. However, instead of using the backtracking method, it does a BFS-style search.

Basically, Branch and Bound involves three different kinds of nodes.

  1. A live node is a created node whose children have not yet been formed.
  2. The offspring of the E-node which is a live node, are now being investigated. Or to put it another way, an E-node is a node that is currently expanding.
  3. A created node which is not to be developed or examined further is referred to as a dead node. A dead node has already extended all of its offspring.

Costs function :

In the search tree, each node Y has a corresponding costs. The next E-node may be found using the costs function. The E-node with the lowest costs is the next one. The function can be defined as :

The optimum costs function for an algorithm for 8 puzzles is :

We suppose that it will costs one unit to move a tile in any direction. In light of this, we create the following costs function for the 8-puzzle algorithm :

There is an algorithm for estimatsing the unknown value of h(y), which is accessible.

Final algorithm :

  • In order to maintain the list of live nodes, algorithm LCSearch employs the functions Least() and Add().
  • Least() identifies a live node with the least c(y), removes it from the list, and returns it.
  • Add(y) adds y to the list of live nodes.
  • Add(y) implements the list of live nodes as a min-heap.

The route taken by the aforementioned algorithm to arrive at the final configuration of the 8-Puzzle from the starting configuration supplied is shown in the diagram below. Keep in mind that only nodes with the lowest costs function value are extended.

8 Puzzle problem in Python

Code :


1 2 3 
5 6 0 
7 8 4 

1 2 3 
5 0 6 
7 8 4 

1 2 3 
5 8 6 
7 0 4 

1 2 3 
5 8 6 
0 7 4

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