A method for the sliding puzzle

Solve it in layers, never all at once

The mistake that makes this puzzle feel impossible is trying to improve the whole board at the same time. Every tile you move disturbs another, so global progress goes round in circles.

Instead, finish regions and never touch them again. The order is: the top row, then the leftmost column, then repeat on what is left.

This is the whole strategy. A four by four becomes a three by three becomes a two by two. Each stage is easier than the last, and nothing you finish ever needs to move again.

Moving one tile without wrecking the rest

To bring a tile somewhere, you do not move the tile. You move the gap, walking it around the board until it sits next to the tile on the side you want the tile to travel towards, then slide the tile into it.

The gap will disturb whatever it walks through, so route it deliberately:

1. Never walk the gap through finished territory. Go the long way round, below or to the right of what you have already solved.
2. Move the target tile along a row or column, not diagonally. There is no diagonal move; a diagonal step is really two moves and it is easy to lose track.
3. Bring the tile to its column first, then up its column. Or the reverse. Picking one order and sticking to it makes the routes predictable.

Counting moves is worth doing while you learn. Our board keeps a running count and remembers your best per size, which is a fair measure of how tidy your gap routing has become.

The last two tiles of a row

Here is the position that stops people. Three tiles of the top row are placed, and the last two — say 3 and 4 — will not both go in. Put 3 in its place and 4 is in the way; fix 4 and 3 pops out.

The answer is to stop trying to place them where they belong. Place them rotated instead:

The same trick finishes the last two tiles of a column, turned ninety degrees: put the upper tile in the bottom position and the other one to its right, then rotate. Once you have done each of them twice they stop needing to be looked up.

The three-tile rotation

Underneath the trick above is the one manoeuvre worth genuinely learning. With the gap in a corner of a two by two area, walking it around that little square cycles the three tiles in it and leaves everything outside untouched.

That is your precision tool. Whenever two tiles are almost right and swapping them would fix everything, the answer is a rotation in a two by two block, not a long journey across the board. It is also the reason the last-two trick works: setting the pair up rotated means one cycle puts both home.

The final two by two

When only a two by two corner is left, there are three tiles and a gap, and the only thing you can do is rotate them. So either they come out in order within three rotations, or they never will.

If they will not, you have made an error earlier — almost always a tile placed in the wrong row during the layer stage. On a board shuffled from the solved state this ending is always reachable, so it is worth undoing rather than restarting.

Three by three, four by four, five by five

The three by three is a good place to practise the rotation, because the whole puzzle is a top row and then a two by two. The four by four is the classic. The five by five is the same method with two extra layers and roughly four times the moves.

On the larger boards, the layer order matters more. Finish the top row and the left column as a pair, working inwards, rather than completing an entire row first — a finished full row leaves the remaining region awkwardly shaped and the gap has fewer safe routes.

If you want a number to chase: a well-routed four by four from a thorough shuffle takes most people somewhere between eighty and a hundred and forty moves. The theoretical worst case is eighty, and getting near that requires planning several tiles ahead rather than the layer method.