How to Solve the 15 Puzzle — A Method That Always Works
By Vladimir, founder of Burmly · Updated June 10, 2026 · 4 min read
The 15 puzzle looks like a game of trial and error, but it has a method that works on every solvable position — no memorization beyond two small tricks. Once you know it, a scrambled board takes a couple of minutes, and most of your moves stop being guesses.
The rules in one paragraph
Fifteen numbered tiles sit in a 4×4 frame with one empty space. A move slides any tile adjacent to the gap into it. The goal: arrange 1–15 in order, reading left to right, top to bottom, with the gap ending bottom-right. That's it — the entire difficulty is in the geometry of getting there.
The method: rows first, then columns
Work top-down, then left-to-right: two full rows first, then the remaining 2×4 block column by column, finishing with a single 2×2 rotation.
The core idea is to solve the board in layers and never touch a finished layer again:
Top row (1, 2, 3, 4). Place 1 and 2 directly. Tiles 3 and 4 need the corner trick described below.
Second row (5, 6, 7, 8). Same procedure, same corner trick for 7 and 8.
Left columns of the remaining 2×4 block. Place 9 and 13 as a pair, then 10 and 14 as a pair, using the column version of the same trick.
The final 2×2. Only 11, 12 and 15 remain with the gap. Rotate the four cells until they fall into place — three moves at most.
The corner trick (finishing a row)
You can't place 3 and then 4 directly — sliding 4 into the corner would push 3 out. Instead:
Target: ... [3][4] at the end of the row
1. Put tile 3 in the CORNER (where 4 belongs)
2. Put tile 4 directly BELOW that corner
3. Slide 3 left into its real cell, then 4 up into the corner
[ · ][ · ][ 3 ] [ · ][ 3 ][ 4 ]
[ · ][ · ][ 4 ] → [ · ][ · ][ · ]
The pair rotates into place in two moves, and the row is finished without disturbing anything you've already solved.
The column trick (the bottom two rows)
After two rows are done, you solve the remaining 2×4 strip column by column with the same rotation, turned on its side. For the 9/13 column:
1. Put tile 13 in the TOP-LEFT cell of the strip (where 9 belongs)
2. Put tile 9 immediately to its RIGHT
3. Slide 13 down into the bottom-left corner, then 9 left into place
[13][ 9 ][ · ][ · ] [ 9 ][ · ][ · ][ · ]
[ · ][ · ][ · ][ · ] → [13][ · ][ · ][ · ]
Repeat for 10 and 14. What's left is a 2×2 with three tiles — cycle them around until 11, 12 and 15 are home.
Why some arrangements can't be solved
Exactly half of all possible tile arrangements are unsolvable — a parity argument proven back in the 1870s, when puzzle maker Sam Loyd famously offered a prize for solving the 14-15 swap (two tiles exchanged, everything else in place). No one ever collected, because no sequence of legal moves can do it. Online versions, including ours, scramble by making legal moves backwards from the solved state, so every position you're given is solvable. If a physical puzzle has been taken apart and reassembled wrong, no method will save it.
Speed tip: Think in tile paths, not single moves. Decide where a tile needs to go, walk the gap around to the far side of it, and push. Counting individual slides is what makes beginners slow — the gap choreography becomes automatic within a few games.
Common mistakes
Solving the rows out of order. The method depends on never revisiting a finished layer. If you place the bottom rows first, the top rows become nearly impossible to fix.
Placing 4 before 3. The corner trick exists precisely because direct placement of the last two tiles in a row deadlocks. Stage them, then rotate.
Moving without a target. Random sliding undoes progress. Every move should be part of a specific tile's path.
Try the method on a real board
Play the 15 puzzle on Burmly — free, no ads, with a move counter and timer.