I have a modified 8 puzzle problem such that each transition's cost is associated with the number of the piece that is moved. So, for example, if piece "3" is moved, the move would cost 3 units.
I am trying to find an admissible heuristic that dominates the Manhattan distance, but am having trouble deriving one.
Since the Manhattan distance gives us the distance of any tile from its end goal, wouldn't a heuristic that dominates the Manhattan distance not be admissible? For example, if a piece is 1 state away from its goal state, how can there be an admissible heuristic that is $>=$ 1 for that piece?