8-puzzle which has the numbers in order but has gap in between If the numbers of the 8-puzzle are all in order, but the blank tile is somewhere in between, is this puzzle solvable? (for example, $\begin{bmatrix} 1 & 0 & 2 \\ 3 & 4 & 5 \\ 6 & 7 & 8 \end{bmatrix}$, where 0 represents the blank tile)

Count how far the gap tile is displaced.  In your case it has been moved up 2 and over 1.  So that is a displacement of 3.  Call that "odd" or 1.
Now do that for every square.
1 is in place.  So that's displaced 0.
2 is over 1 so that's displaced 1.
3 is down one and over 3 so that's 3. odd. 1.
4 is over 1. 1.
5 is over 1. 1.
6 is displaced 3. odd. 1.
7 is displace 1.
8 is displaced 1.
Add those all up.  8.  even.  0.
Even parity is solvable.
Odd parity is not.
$\begin{bmatrix} 1 & 0 & 2 \\ 3 & 4 & 5 \\ 6 & 7 & 8 \end{bmatrix}$
$\begin{bmatrix} 1 & 2 & 5 \\ 0 & 3 & 4 \\ 6 & 7 & 8 \end{bmatrix}$
$\begin{bmatrix} 2 & 3 & 5 \\ 1 & 0 & 4 \\ 6 & 7 & 8 \end{bmatrix}$
$\begin{bmatrix} 1 & 2 &3 \\ 4 & 5 & 0 \\ 6 & 7 & 8 \end{bmatrix}$
$\begin{bmatrix} 1 & 2 &3 \\ 4 & 5 & 8 \\ 0 & 6 & 7 \end{bmatrix}$
$\begin{bmatrix} 1 & 2 &3 \\ 5 & 0 & 8 \\ 4 & 6 & 7 \end{bmatrix}$
$\begin{bmatrix} 1 & 2 &3 \\ 5 & 6 & 8 \\ 4 & 0 & 7 \end{bmatrix}$
$\begin{bmatrix} 1 & 2 &3 \\ 5 & 6 & 0 \\ 4 & 7 & 8 \end{bmatrix}$
$\begin{bmatrix} 1 & 2 &3 \\ 4 & 5 & 6 \\ 0 & 7 & 8 \end{bmatrix}$
$\begin{bmatrix} 1 & 2 &3 \\ 4 & 5 & 6 \\  7 & 8 & 0 \end{bmatrix}$
