Guide · works on any board size
One method, five steps, any size. Practise it on the board below as you read.
Every sliding puzzle, from the 3×3 8 puzzle to a 6×6 board, is solved the same way: in layers, from the outside in. You finish the top row and lock it. You finish the left column and lock it. What is left is a smaller puzzle, and you repeat until only a 2×2 corner remains.
The method is slower than an optimal solution, usually two to four times as many moves, but it never gets stuck and it needs no memorisation beyond one short sequence for the awkward last two tiles of each line.
Find tile 1 and bring it to the top-left corner. Move the empty space next to the tile you want to move, then slide the tile into it; the empty space is the thing you steer, the tiles just follow.
Continue with 2, then 3, and so on, until only the last two tiles of the row are missing. On a 4×4 board that means 1 and 2 are placed and 3 and 4 remain. Tiles already in place must not move again.
The last two tiles of a row cannot be placed one at a time; the second one has nowhere to enter. Instead, put the last tile (4) where the second-to-last belongs, and park the second-to-last tile (3) directly below it.
Now slide 4 to the right into the corner, and slide 3 up into the gap it left. Both are in place. If 3 happens to be stuck in the corner where 4 needs to go, move it out first: bring 3 down two rows, position 4 and 3 as described, and repeat.
Turn the board ninety degrees in your head. The left column is now your row. Place its tiles from the top down, again leaving the last two for the trick: park the last tile where the second-to-last belongs, park the second-to-last to its right, then slide down and slide left.
On a 4×4 board, after this step tiles 1, 2, 3, 4, 5, 9 and 13 are locked and a 3×3 puzzle remains in the bottom-right.
Treat what is left as a fresh puzzle. Solve its top row (6, 7, 8 on a 4×4), using the two-tile trick for 7 and 8. Solve its left column (10, 14), using the trick again. Never touch a locked tile.
Each pass removes one row and one column. A 5×5 needs two passes before the endgame, a 6×6 three.
You are left with three tiles and the empty space in a 2×2 square. Only one kind of move is possible now: rotating the three tiles clockwise or anticlockwise around the square. Rotate until they read in order with the empty space at bottom right.
If they cycle forever without lining up, the puzzle is unsolvable; it was assembled with two tiles swapped. A board shuffled by sliding, like every board on this site, always finishes here.
A 4×4 board. Solve the top row first, then the left column, and watch the puzzle shrink.
For a human, the row-by-row method above: complete the top row, then the left column, and repeat on the remaining smaller board until a 2×2 corner is left. It takes 80 to 150 moves on a 4×4 board and about a minute of practice to learn. Optimal solutions are much shorter, 53 moves on average for a 4×4, but finding them requires a computer search; the solver on this site does exactly that for 3×3 and 4×4.
Place the higher-numbered tile where the lower one belongs, park the lower tile directly below it, then slide the higher tile sideways into the corner and the lower tile up into the gap. For example on a 4×4: put 4 in the third cell of the top row, put 3 below it in the second row, slide 4 right, slide 3 up. The same trick, rotated, finishes each column.
If you have reached a 2×2 corner where the last three tiles cycle but never line up, the puzzle is unsolvable. Exactly half of all arrangements are unreachable by sliding, and a physical puzzle that was disassembled or had tiles popped out often lands in that half. Swap any two adjacent tiles by hand and it becomes solvable.
Yes, identically. Solve the top row and left column, and a 5×5 becomes a 4×4, then a 3×3, then a 2×2. The only difference is length: a 5×5 solve by hand is typically 150 to 300 moves, a 6×6 300 to 600.