# How many configurations of 8-puzzle are there?

## How many configurations of 8-puzzle are there?

The classical 8-puzzle belongs to the family of sliding blocks. My book (Artificial intelligence A modern approach by Stuart Russell and peter Norwig) says that the 8-puzzle has 9!/2 possible states.

### What is the 8-puzzle problem?

The 8-puzzle problem is a puzzle invented and popularized by Noyes Palmer Chapman in the 1870s. It is played on a 3-by-3 grid with 8 square blocks labeled 1 through 8 and a blank square. Your goal is to rearrange the blocks so that they are in order.

#### Why are the 8 puzzle States divided into two disjoint sets?

We have to choose one of the goal states to be reached because only one of them is reachable from the given initial state. Sounds interesting? Lets see why the 8-puzzle states are divided into two disjoint sets, such that no state in one set can be transformed into a state in the other set by any number of moves.

**Why are there two possible arrangements in the 8 puzzle problem?**

As it’s mentioned in the 8 puzzle problem page, the game has two possible arrangements. We have to choose one of the goal states to be reached because only one of them is reachable from the given initial state. Sounds interesting?

**When is a collection of more than two sets called disjoint?**

A collection of more than two sets is called disjoint if any two distinct sets of the collection are disjoint. . Alternatively, some authors use the term disjoint to refer to this notion as well. (every two distinct sets in the family are disjoint).

## How to solve the 8 puzzle using branch and bound?

Given a 3×3 board with 8 tiles (every tile has one number from 1 to 8) and one empty space. The objective is to place the numbers on tiles to match the final configuration using the empty space. We can slide four adjacent (left, right, above, and below) tiles into the empty space. 1. DFS (Brute-Force)