Write a program to solve the 8-puzzle problem (and its natural generalizations) using the A* search algorithm.

The problem. The 8-puzzle is a sliding puzzle that is played on a 3-by-3 grid with 8 square tiles labeled 1 through 8, plus a blank square. The goal is to rearrange the tiles so that they are in row-major order, using as few moves as possible. You are permitted to slide tiles either horizontally or vertically into the blank square. The following diagram shows a sequence of moves from an initial board (left) to the goal board (right).

8puzzle 4 moves

Board data type. To begin, create a data type that models an n-by-n board with sliding tiles. Implement an immutable data type Board with the following API:

public class Board {

    // create a board from an n-by-n array of tiles,
    // where tiles[row][col] = tile at (row, col)
    public Board(int[][] tiles)
    // string representation of this board
    public String toString()

    // tile at (row, col) or 0 if blank
    public int tileAt(int row, int col)

    // board size n
    public int size()

    // number of tiles out of place
    public int hamming()

    // sum of Manhattan distances between tiles and goal
    public int manhattan()

    // is this board the goal board?
    public boolean isGoal()

    // does this board equal y?
    public boolean equals(Object y)

    // all neighboring boards
    public Iterable<Board> neighbors()

    // is this board solvable?
    public boolean isSolvable()

    // unit testing (required)
    public static void main(String[] args)


Constructor.  You may assume that the constructor receives an n-by-n array containing a permutation of the n2 integers between 0 and n2 − 1, where 0 represents the blank square. You may also assume that 2 ≤ n ≤ 32,768.

String representation.  The toString() method returns a string composed of n + 1 lines. The first line contains the board size n; the remaining n lines contains the n-by-n grid of tiles in row-major order, using 0 to designate the blank square.

String representation

Tile extraction.  Throw a java.lang.IllegalArgumentException in tileAt() unless both row and col are between 0 and n − 1.

Hamming and Manhattan distances.  To measure how close a board is to the goal board, we define two notions of distance. The Hamming distance betweeen a board and the goal board is the number of tiles in the wrong position. The Manhattan distance between a board and the goal board is the sum of the Manhattan distances (sum of the vertical and horizontal distance) from the tiles to their goal positions.

Hamming and Manhattan distances

Comparing two boards for equality.  Two boards are equal if they are have the same size and their corresponding tiles are in the same positions. The equals() method is inherited from java.lang.Object, so it must obey all of Java’s requirements.

Neighboring boards.  The neighbors() method returns an iterable containing the neighbors of the board. Depending on the location of the blank square, a board can have 2, 3, or 4 neighbors.

Neighboring boards

Detecting unsolvable boards.  An efficient approach for detecting unsolvable boards is described in the next section.

Unit testing.  Your main() method must call each public method directly and help verify that they works as prescribed (e.g., by printing results to standard output).

Performance requirements.  In the worst case, your implementation must support size(), tileAt(), isGoal(), hamming(), and manhattan() in constant time; the constructor, equals(), toString(), and neighbors() in time proportional to n² (or better); and isSolvable() in time proportional to n4 (or better). See the optimizations section for information on making hamming() and manhattan() take constant time.

Detecting unsolvable boards. Not all initial boards can lead to the goal board by a sequence of moves, including these two:

unsolvable slider puzzles

Remarkably, we can determine whether a board is solvable without solving it! To do so, we count inversions, as described next.

A* search. Now, we describe a solution to the 8-puzzle problem that illustrates a general artificial intelligence methodology known as the A* search algorithm. We define a search node of the game to be a board, the number of moves made to reach the board, and the previous search node. First, insert the initial search node (the initial board, 0 moves, and a null previous search node) into a priority queue. Then, delete from the priority queue the search node with the minimum priority, and insert onto the priority queue all neighboring search nodes (those that can be reached in one move from the dequeued search node). Repeat this procedure until the search node dequeued corresponds to the goal board.

The efficacy of this approach hinges on the choice of priority function for a search node. We consider two priority functions:

To solve the puzzle from a given search node on the priority queue, the total number of moves we need to make (including those already made) is at least its priority, using either the Hamming or Manhattan priority function. Why? Consequently, when the goal board is dequeued, we have discovered not only a sequence of moves from the initial board to the goal board, but one that makes the fewest moves. (Challenge for the mathematically inclined: prove this fact.)

Game tree. One way to view the computation is as a game tree, where each search node is a node in the game tree and the children of a node correspond to its neighboring search nodes. The root of the game tree is the initial search node; the internal nodes have already been processed; the leaf nodes are maintained in a priority queue; at each step, the A* algorithm removes the node with the smallest priority from the priority queue and processes it (by adding its children to both the game tree and the priority queue).

For example, the following diagram illustrates the game tree after each of the first three steps of running the A* search algorithm on a 3-by-3 puzzle using the Manhattan priority function.

8puzzle game tree

Solver data type. In this part, you will implement A* search to solve n-by-n slider puzzles. Create an immutable data type Solver with the following API:

public class Solver {

    // find a solution to the initial board (using the A* algorithm)
    public Solver(Board initial)

    // min number of moves to solve initial board
    public int moves()

    // sequence of boards in a shortest solution
    public Iterable<Board> solution()

    // test client (see below) 
    public static void main(String[] args)


Implementation requirement.  To implement the A* algorithm, you must use the MinPQ data type for the priority queue.

Corner case.  Throw a java.lang.IllegalArgumentException in the constructor if the argument is null.

Unsolvable boards.  Throw a java.lang.IllegalArgumentException in the constructor if the initial board is not solvable.

Test client. Your test client should take the name of an input file as a command-line argument and print the minimum number of moves to solve the puzzle and a corresponding solution. The input file contains the board size n, followed by the n-by-n grid of tiles, using 0 to designate the blank square.

~/Desktop/8puzzle> more puzzle04.txt
 0  1  3
 4  2  5
 7  8  6

~/Desktop/8puzzle> java-algs4 Solver puzzle04.txt
Minimum number of moves = 4
 0  1  3 
 4  2  5 
 7  8  6 

 1  0  3 
 4  2  5 
 7  8  6 

 1  2  3 
 4  0  5 
 7  8  6 

 1  2  3 
 4  5  0   
 7  8  6 

 1  2  3 
 4  5  6 
 7  8  0
~/Desktop/8puzzle> more puzzle3x3-unsolvable.txt
 1  2  3
 4  5  6
 8  7  0

Unsolvable puzzle

Two optimizations. To speed up your solver, implement the following two optimizations:

Challenge for the bored. Implement a better solution which is capable of solving puzzles that the required solution is incapable of solving.

Deliverables. Submit the files Board.java and Solver.java (with the Manhattan priority). We will supply algs4.jar. Your may not call any library functions other than those in java.lang, java.util, and algs4.jar. You must use MinPQ for the priority queue. Finally, submit a readme.txt file and answer the questions.


file points
Board.java 15
Solver.java 20
readme.txt 5

Reminder: You can lose up to 4 points for poor style and up to 4 points for inadequate unit testing.