This class shows how to determine if a 3x3 sliding puzzle is solvable or not.

Solvable.java

/**
 * This class represents how to determine the solvability
 * of a 3 x 3 sliding puzzle.
 *
 * @author Casey Scarborough
 */
public class Solvable {
  
  public static void main(String[] args) {

    // These puzzles are known solvable.
    int[] a = { 0, 1, 2, 4, 5, 3, 7, 8, 6 };
    int[] b = { 1, 2, 3, 0, 4, 6, 7, 5, 8 };
    int[] c = { 1, 0, 3, 7, 2, 5, 8, 4, 6 };

    // These are known not solvable.
    int[] d = { 1, 2, 3, 6, 8, 4, 5, 7, 0 };
    int[] e = { 1, 2, 3, 4, 5, 6, 8, 7, 0 };
    int[] f = { 1, 5, 0, 3, 2, 8, 4, 6, 7 };
    
    if (isSolvable(a) && isSolvable(b) && isSolvable(c))
      System.out.println("These puzzles are solvable."); 

    if (!isSolvable(d) && !isSolvable(e) && !isSolvable(f)) 
      System.out.println("These puzzles are not solvable."); 

  }

  // This method takes a two dimensional array representing
  // a sliding puzzle, and determines if it is solvable.
  public static boolean isSolvable(int[] p) {
    int inversions = 0;
    
    for(int i = 0; i < p.length - 1; i++) {
      // Check if a larger number exists after the current
      // place in the array, if so increment inversions.
      for(int j = i + 1; j < p.length; j++)
        if(p[i] > p[j]) inversions++;

      // Determine if the distance of the blank space from the bottom 
      // right is even or odd, and increment inversions if it is odd.
      if(p[i] == 0 && i % 2 == 1) inversions++;
    }

    // If inversions is even, the puzzle is solvable.
    return (inversions % 2 == 0);
  }
}