Admissible Heuristic for n-puzzle if multiple tiles can be slided at once

Question

In the context of n-puzzle, consider a variation that we can slide multiple tiles in one row or column, if they are touching each other.  The cost of this slide operation is the same as that of sliding one tile.

Design an admissible and useful heuristic for this problem.

Answer 1

for any tile, it can swap with the empty tile，and each swap counts for max(|y1-y2|,|x1-x2|) moves, and if they are in the same row or same col, this move will count for 0. every time, if the empty tile is not in the right place(the tile the empty tile should be in goal position), then swap it with the number should be in this tile in the goal position(to make this number matched). If the empty tile is in goal position, then swap it with the farthest mismatched number, do until to the goal position. And heuristic is the sum of moves.