If the initial position has only short-range pieces, then at each stage each player has only finitely many possible moves, and so the game tree is finitely branching. Thus, given an initial configuration, one may simply compute the entire game tree to depth $n$ (which will be a finite tree) and use back-propagation to determine the winning strategy and whether there is one for that position. Thus, we may compute the answer to the mate-in-n problem for such a position and furthermore compute the moves of a winning strategy. 

The subtle difficulty of the mate-in-n problem arises only when there are long-range pieces, since in this case, the tree becomes infinitely branching, and so one cannot search the tree to depth $n$ in finite time. Nevertheless, as we explain in the article, the mate-in-n problem is still decidable by other means than searching through the tree.