I chanced upon this puzzle in this question on the Anime & Manga site, and, like the OP, tried to solve it without any success. Here is a representation of the puzzle: the blocks may only be moved forward or backward along their shorter sides, and we must get the block labelled with a black X to the exit.
I then tried to prove that the puzzle wasn't solvable using the following framework:
Argue that at some time the block we must move to the exit will be exactly one space away from it, and that moreover, we will only need to move the other blocks at most once to remove any obstructions to the exit.
Use proof by contradiction and the layout of the puzzle to argue that none of the possible configurations for which the block is at the space I specify are actually at most a single move away from making the exit.
I ran into a brick wall with this: there were four possible cases to consider (based on the placement of (IV)), and while I could easily apply (2) to two of them, I didn't have nearly as much success with the other two.
I was wondering if I simply should have chosen a different position to work from, but it occurred to me that unless I had extremely lucky intuition, it would be difficult to make arguments from anything other than the penultimate placement of the blocks. (Naturally, it also occurred to me that I could probably just check for solvability by writing a computer program to check for all possible configurations of this puzzle, but my programming skills are iffy, and I found such a "brute-force" method too uninsightful.) Moreover, my proof-strategy would only have worked if the puzzle were indeed unsolvable.
I don't suppose that there is necessarily a very intuitive generalization of finding the solvability of such puzzles, but at the very least: what would be a good strategy for me to use to determine whether or not this puzzle can be solved, without too much brute-force?