There is a very clear analysis of the 4x4 case here, with enough information to give you some good hints on tackling the NxN case.
Consider an sliding jigsaw (SJ) with an empty block and numbered blocks from to (see the attachment for a configuration of a SJ, where blocks or can move to the empty block, leaving their blocks empty, which any of their adjacent neighbors can in turn move to, ect.)
Definition 1: A configuration is standard, if the empty block is at the northwest corner of the board.
Definition 2: A configuration is ordered, if every block is greater than the block to its left (the first block of a row is greater than the last block of the previous row), ignoring the empty block.
Question 1: Starting from a standard configuration of an SJ, how many different standard configurations can we reach?
Question 2: Starting from an arbitary configuration, can we reach an ordered SJ?