This is a procedural guide. Print it out, or read it next to a fresh 15 puzzle on your phone, and follow the steps. By the end you will have a method that solves any 15 puzzle in three to seven minutes.
What you are aiming at
The 4×4 board with tiles in order:
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 _
The empty cell is in the bottom-right.
Step 1 — solve the top row (1, 2, 3, 4)
Place tile 1 in the top-left first. This is easy — slide tiles around until 1 lands in position. Don't worry about disturbing anything else, because nothing is in place yet.
Place tile 2 to the right of 1. Same approach: bring 2 toward the top, then nudge it next to 1.
Now you reach the tricky part: tiles 3 and 4 need to go into the top-right of the row, but you cannot just place 4 there — when 4 sits in the top-right, you usually need 3 to follow, and getting 3 to slide in usually displaces 4.
The technique:
- Place 4 in the top-right corner first.
- Place 3 directly under 4 (in position 8, the second-row-right cell).
- Now rotate the 4-3 pair clockwise into the corner. This means: move the empty cell to the right of 4 (position 4, the top-right itself), slide 4 left, slide 3 up, slide 4 right.
After this rotation, both 3 and 4 are in place and the rest of the board has only been slightly disturbed. The top row is locked.
(This rotation is the L-shaped corner manoeuvre. Memorise it — you will use it on every row corner and every column corner of every puzzle for the rest of your life.)
Step 2 — solve the left column (5, 9, 13)
Don't disturb the top row. From now on, you only work with the bottom three rows.
Place tile 5 in position 5 (top of the left column, just below 1). This is usually one or two slides.
Place tile 9 below 5. Same approach.
Place tiles 13 and ... wait, you have only three left-column tiles to do (5, 9, 13) — the bottom-left corner is 13. Use the same L-shaped corner manoeuvre, mirrored vertically:
- Place 13 in position 9 (the cell where 9 was — wait, you already placed 9 there).
- Actually the standard move: place 9 in the bottom-left corner first, place 13 directly to its right, then rotate the pair clockwise into place: empty above 9, slide 9 up, slide 13 left, slide 9 down.
After this, the top row and left column are locked. You are working on a 3×3 subpuzzle.
Step 3 — solve the 3×3 endgame
What remains is a 3×3 grid (rows 2-3-4, columns 2-3-4) holding tiles 6, 7, 8, 10, 11, 12, 14, 15 and the empty. The goal positions inside this sub-grid are:
6 7 8
10 11 12
14 15 _
This is now an 8 puzzle, with a different goal state but the same rules. Apply the same row-and-column method recursively:
- Solve the top of the sub-grid (tiles 6, 7, 8). Place 6 first. Then the 7-8 corner uses the L-rotation again.
- The bottom 2×2 holds 10, 11, 14, 15. Three tiles and the empty. You can usually rotate them into place by repeated single moves.
If the bottom 2×2 ends up with two tiles swapped, that means the original puzzle is unsolvable — which only happens if the app generated it badly, not by chance from a random shuffle followed by parity checking.
Common stuck points
"I have everything but tiles 3 and 4 are reversed." This is the corner-rotation trick. Move them both out of the corner together, then rotate them back in the correct orientation.
"I'm trying to place a tile but every move messes up what I already placed." You are inside a row or column you locked. Step back and verify which tiles are "supposed to stay still" — once a row or column is locked, you can only work in the unlocked area below or to the right.
"My puzzle won't finish — the last two tiles are swapped." Two cases: either you accidentally moved a locked tile (rare), or the puzzle is unsolvable (also rare). Test by resetting and using the parity check.
How fast can you go
A first-time solver, applying this method literally, takes 15-25 minutes. By the third solve, the same method takes 8 minutes. By the tenth, 3-5 minutes. Speed-solvers under two minutes use the same method with motor memory.
There is no faster human method. The row-and-column reduction is provably the cleanest hand technique known. Computer optimal solutions are shorter (~52 moves for a random board) but require search algorithms; a human implementing them would need to plan twenty-plus moves ahead, which is impractical.
What's next
Once the 4×4 feels routine, the same method works on the 24 puzzle (5×5) and the 35 puzzle (6×6) without modification — just more rows and columns to peel before you reach the embedded 3×3.
If you want the general method without the 4×4 specifics, see how to solve a slide puzzle. If you want the simplest possible worked example, see 3×3 slide puzzle solution.