# How to check if a 8-puzzle is solvable?

I have a 8-puzzle

1|2|3
-+-+-
4|5|6
-+-+-
|8|7


How can be checked if the puzzle is solvable?

Wikipedia states that it is solvable, but does not prove it. Can anybody explain the prove?

-
What is a $9$ puzzle? – Michael Albanese Feb 3 at 11:01
It is a 15-puzzle with only 9 items. – ceving Feb 3 at 11:02
Ok it would be more precise to call it 8-puzzle. – ceving Feb 3 at 11:07
Where does Wikipedia claim this is solvable? – Chris Eagle Feb 3 at 11:17
A careful read of the Wikipedia article shows that what they're claiming is that if it's solvable then no more than 31 single tile moves required. Before that it definitely says the $n$ puzzle is only solvable for even permutations. – coffeemath Feb 3 at 14:29
If you ignore the gap and just look at the ordered sequence of numbers, any "horizontal" move leaves the sequence unchanged and any "vertical" move of the puzzle has the form $$(\ldots, x, y, z, \ldots)\to (\ldots, y, z, x, \ldots)$$ or vice versa. These are even permutations and therefore the group of possible permutations is a subgroup of $A_8$ (alternating group) and cannot be all of $S_8$ (full symmetric group). The situation in your post corresponds to an odd permutation of the target ordering and therefore is not solvable.