Here is a simple rule for telling if an 8 puzzle is
solvable.

Recall that our goal state is
  
  1 2 3 
  4 5 6
  7 8 0

where the 0 represents the blank.

Now, consider the following arrangement of tiles

  1 2 3     
  4 5 7
  6 8 0

The first line is in the correct order. The second
line is correct except that counter 7 comes before
counter 6. We can call this violation of the order
an "inversion".

If the number of inversions is even then the puzzle
is solvable. If the number of inversions is odd 
then the puzzle is unsolvable.

Consider the puzzle

2 4 3
1 0 6
7 5 8

We have the following inversions

  2 comes before 1
  4 comes before 3 and 1
  3 comes before 1
  6 comes before 5
  7 comes before 5

for a total of 6 inversions. This puzzle is solvable.
Note that we don't need to consider the blank (0).


More examples:

  1 5 8
  0 2 3
  4 6 7
 
  5 comes before 2 3 4
  8 comes before 2 3 4 6 7
  
For a total of 8 inversions. This puzzle is solvable.



  5 1 8
  0 2 3
  4 6 7

  5 comes before 1 2 3 4
  8 comes before 2 3 4 6 7
 
for a total of 9 inversions. This puzzle is not solvable.



  2 0 7
  8 5 4
  3 6 1

  2 comes before 1
  7 comes before 5 4 3 6 1
  8 comes before 5 4 3 6 1
  5 comes before 4 3 1
  4 comes before 3 1
  3 comes before 1
  6 comes before 1

for a total of 18 inversions. This puzzle is solvable



  8 7 6
  5 4 3
  2 1 0

  8 comes before 7 6 5 4 3 2 1
  7 comes before 6 5 4 3 2 1
  6 comes before 5 4 3 2 1
  5 comes before 4 3 2 1
  4 comes before 3 2 1
  3 comes before 2 1
  2 comes before 1

for a total of 28 inversions. This puzzle is solvable (although
you may run out of memory before finding a solution).