For which permutations $\sigma \in S_{n^2-1}$

is the n x n slide puzzle

$$<br /> \begin{matrix}<br /> \sigma(1) &amp; \sigma(2) &amp; \cdots &amp; \sigma(n-1) &amp; \sigma(n)\\<br /> \sigma(n+1) &amp; \sigma(n+2) &amp; \cdots &amp; \sigma(2n-1) &amp; \sigma(2n)\\<br /> &amp; \vdots &amp; &amp; &amp; \\<br /> \sigma(n^2-n+1) &amp; \sigma(n^2-n+2) &amp; \cdots &amp; \sigma(n^2-1) &amp; \bullet<br /> \end{matrix}<br />$$

solvable? (By solvable this means get to

$$<br /> \begin{matrix}<br /> 1 &amp; 2 &amp; \cdots &amp; n-1 &amp; n \\<br /> n+1 &amp; n+2 &amp; \cdots &amp; 2n-1 &amp; 2n \\<br /> &amp; \vdots &amp; &amp; &amp; \\<br /> n^2-n+1 &amp; n^2-n+2 &amp; \cdots &amp; n^2-1 &amp; \bullet<br /> \end{matrix}<br />$$

where the only allowed moves are transposing the empty slot $\bullet$ with a horizontally- or vertically-adjacent slot.)

Tbh I'm not sure what the full answer is.

Hint to get started: odd permutations are not solvable.