invariant

The invariant is the [|parity of permutations] of all 16 squares (15 pieces plus empty square) plus the parity of the [|taxicab distance] moved by the empty square. This is an invariant because each move changes the parity of the permutation and the parity of the taxicab distance. In particular if the empty square is not moved the permutation of the remaining pieces must be even. The same argument works for all rectangular boards, or more generally for all boards with no odd cycles.