The 15-puzzle with two tiles swapped

Idea. Find an invariant that mixes the parity of inversions with the row of the blank.

Invariant. For a position, let $N$ be the number of inversions of the sequence of the $15$ tiles read row by row, and let $r\in\{1,2,3,4\}$ be the row of the empty cell from the top. The parity of $N+r$ does not change under a move.
- Horizontal move. $N$ and $r$ are unchanged.
- Vertical move. $r$ changes by $\pm1$. The moved tile jumps over exactly $3$ other tiles, so $N$ changes by an odd number. Hence $N+r$ keeps its parity.

Conclusion. The solved position has $N=0$ and $r=4$, so $N+r$ is even. The position with tiles $14,15$ swapped has $N=1$ and $r=4$, so $N+r$ is odd. Since moves preserve the parity, this position cannot be reached. ∎

Remark. Sam Loyd's famous unsolvable puzzle. Exactly half of all positions are reachable.

Lesson. Parity of inversions is the invariant of sliding puzzles; correct it with the position of the blank.