Tricks to solve the nxn puzzle

Here are a few practical tricks to help you to solve the puzzle.

How to get the wrong tile out of the corner

Typically the situation is such that the wrong tile in the corner is surrounded by "right" tiles — on the wrong tile's side there is the tile that shall be there and below the wrong tile is the tile that should be in the corner. Since the wrong tile cannot jump over the other tiles, the right tiles must make room for the wrong one to come out of the corner.

First the tile that is on the side of the wrong tile must move sideways away from the wrong tile. If it goes down and the wrong tile comes on top of it, the situation remains blocked. So, the right tiles move sideways away from the wrong tile and in order to make room for this movement, one of the right tiles further afield must move down — not the tile right next to the wrong tile.

This way the wrong tile can move sideways away from the corner and the right tile that was below the corner can take its place in the corner. There will be another wrong tile below the wrong tile that is now one step away from the corner but still in the same row. So the other tile below can be moved away to give the wrong tile room to move down from the row where it was. Now there is room for the right tiles to come back to where they were before they made room for the wrong tile to come out of the corner.

Note that on higher levels this same technique can be used in the middle of a row. Let's imagine a large puzzle where the row we are working on is ready except for one number somewhere near the middle of the row. The right number is standing one row below — just under where it should be. And on its place on the upper row is some tile that doesn't belong to the row. Now this wrong tile is surrounded by right tiles. In order to resolve the situation, use the same technique as when getting the wrong tile out of the corner.

Move the tiles as a "worm"

This trick has to do with efficiency. Especially on larger puzzles you can save many moves by moving a "worm" of tiles across many rows or columns in order to bring them to where they belong.

If we move one tile 5 steps in one direction, how many moves do we need to make? For every step we want our tile to take, we need to relocate the empty tile in front of it before our tile can move. And after every step our tile has taken, the empty tile will be behind it. It takes 4 moves to move the empty tile from behind our tile to front of it. So, for 5 steps we need to use additional 20 moves just to relocate the empty tile (assuming the empty tile needs to be relocated like this also for the first step our tile is going to take). All in all we need to make 25 moves.

The 4 moves that are needed for the relocation of the empty tile are: 1 move to the side, 2 moves to get pass the tile we wish to move, and 1 move sideways to get the empty tile in front of our tile. What if we had a "worm" of 2 tiles we want to move? The relocation of the empty tile would take just 5 moves: 1 aside, 3 to pass, 1 aside. We needed 4 extra moves for 1 tile and 5 extra moves for 2 tiles. Clearly it's more efficient to move 2 tiles as a "worm" than moving first one of them alone and then the second one alone.

For each additional tile in the "worm" the relocation of the empty tile takes just 1 additional move, so the longer the worm, the more efficient the movement. This is due to the fact that the 2 moves aside that the empty tile needs to take are constant no matter how many tiles we have in our worm formation.