How to Solve the 15 Puzzle (Sliding Tile Puzzle) Step by Step

The 15 puzzle is the classic sliding tile game: fifteen numbered tiles in a 4×4 frame with one empty space. You slide tiles into the empty space until they are in order from 1 to 15, with the empty space in the bottom-right corner.

It can feel random at first, but there is a simple step-by-step method that solves it every time.

The goal

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 _

The method in three stages

1. Solve the top row (1, 2, 3, 4).
2. Solve the second row (5, 6, 7, 8).
3. Solve the bottom two rows column by column: first 9 and 13, then 10 and 14, then 11, 12 and 15.

Once a row is done, you never need to touch it again. Each stage is a smaller version of the same puzzle.

Stage 1: The top row

Tiles 1 and 2

Move tile 1 into the top-left corner, then tile 2 next to it. With most of the board free, this is usually easy: move the empty space next to the tile you want, then slide the tile one step at a time toward its target, circling the empty space around it.

Tiles 3 and 4: the tricky part

You cannot simply put 3 in place and then 4, because placing 4 in the corner would require moving 3 out again. Instead:

1. Place 4 in the spot where 3 belongs (third position of the top row).
2. Place 3 directly below it (second row, third column).
3. Move the empty space to the top-right corner.
4. Slide 4 right into the corner, then slide 3 up into place.

The two tiles "rotate" into position together.

If 3 and 4 get stuck in the wrong order, move them both out of the top row and try again. It takes a few extra moves but always works.

Stage 2: The second row

Repeat exactly the same process for 5, 6, 7 and 8. Place 5 and 6 directly, then use the same trick for 7 and 8: put 8 where 7 belongs, 7 below it, and rotate them in together.

Stage 3: The bottom two rows

Now you only have a 2×4 area left. Work in columns from left to right.

Tiles 9 and 13

This is like the 3-and-4 trick turned on its side:

1. Place 13 where 9 belongs (the left end of the third row).
2. Place 9 directly to its right.
3. Move the empty space to the bottom-left corner.
4. Slide 13 down into the corner, then slide 9 left into place.

Tiles 10 and 14

Use the same column trick: 14 where 10 belongs, 10 to its right, then rotate them in.

Tiles 11, 12 and 15

With only a 2×2 area left, circle the three tiles around the empty space until they fall into order. This always works if the puzzle was solvable to begin with.

Why some puzzles cannot be solved

Exactly half of all possible arrangements of the 15 tiles cannot be solved. If you physically pop two tiles out of a real puzzle and swap them, you may create one of these. The most famous example is the original puzzle with 14 and 15 swapped, which became a popular challenge in the 1800s. It has no solution.

Good news: if a game shuffles the puzzle by sliding tiles (rather than placing them randomly), it is always solvable, because sliding can only lead to solvable positions.

Tips

• Think in terms of moving the empty space, not the tiles. To move a tile in a direction, first bring the space next to it on that side.
• Do not disturb finished rows. If a move would require it, look for another route.
• Speed comes later. Learn the method first; shortcuts appear naturally as you practice.

Once you have solved it a few times with this method, the 15 puzzle stops being frustrating and becomes a quick, satisfying routine.