There is no list of fifty techniques for solving slide puzzles by hand. There are essentially four — and once you have all four, board size doesn't matter. This is unusual among logic-puzzle families (sudoku has dozens of distinct techniques) and is part of what makes slide puzzles tractable for new solvers.
This article catalogues the four techniques and shows where each applies.
Technique 1 — Row-and-column reduction
The high-level strategy. Solve the top row, then the left column, then recurse into the smaller subpuzzle. Eventually you reach a 3×3 endgame.
Why this works: solving a row plus a column reduces an N×N puzzle to an (N−1)×(N−1) puzzle of the same kind. The recursion bottoms out at 3×3, which has a small fixed pattern set.
Where to apply: every puzzle of every size. This is the master plan.
The full mechanics are written up in how to solve a slide puzzle.
Technique 2 — The L-shaped corner manoeuvre
The trick for placing the last tile of a row (or the last tile of a column). Without it, you cannot get tile N into the top-right corner of row 1 without disturbing tiles 1...N-1.
The procedure for the top-right corner of row 1:
- Place tile N−1 in the corner (not in its goal position N−1).
- Place tile N directly below N−1 (one row down from the corner).
- Rotate the pair clockwise into place: empty into the corner cell, slide N−1 left, slide N up, slide N−1 right.
Result: tiles N−1 and N are in their goal positions. The cells underneath them may be slightly disturbed but the row is locked.
The bottom-left corner of a column uses the mirrored procedure (counter-clockwise).
Where to apply: at every row corner and every column corner of every size. The single most important technique in slide-puzzle solving.
Technique 3 — The empty-cell routing rule
When you need to move the empty cell from one part of the board to another, you have to route it around already-placed tiles, never through them.
The rule: once a tile is placed, the empty cell may not pass through its position.
In practice, this means you sometimes need to take an indirect route. Want to move the empty from the bottom-right to the top-left, but row 1 is already placed? Route via the bottom-left corner instead.
This is mostly automatic once you have a few hours of practice. New solvers sometimes don't notice when they've broken the rule and accidentally disturbed a placed tile. Drawing the path in advance helps.
Where to apply: continuously, whenever you need to relocate the empty cell across a region with placed tiles.
Technique 4 — The 3×3 endgame
After reducing an N×N puzzle to a 3×3, the remaining puzzle has at most 8 distinct configurations from any "almost solved" state. They can all be cycled into place by sequences of 6–15 single moves.
The endgame has three sub-cases:
Already solved or one cycle away. Solve directly.
Two tiles in the row/column need to swap, three tiles need to cycle. Standard 3-cycle. The technique: choose any tile, displace the two it should swap with, rotate them around it, then put the chosen tile back. About 12 moves.
Two specific tiles are swapped and nothing else. This indicates the original starting position was unsolvable. It cannot arise from a correctly-shuffled puzzle.
Where to apply: at the very end of every solve. The endgame is short — usually under 30 seconds even for a new solver.
Why these four are sufficient
A proof sketch:
- The master plan (technique 1) reduces every puzzle to a 3×3 endgame after finitely many row-and-column placements.
- Each row-and-column placement is a sequence of tile placements: easy interior tiles, then the corner manoeuvre (technique 2) for the last two.
- Each tile placement uses interior moves; routing the empty cell among the unplaced tiles uses technique 3.
- The 3×3 endgame (technique 4) handles the base case.
Cover all four cases and you have a complete algorithm. There is no slide-puzzle position that requires a fifth technique.
This is provably true. It is also why slide puzzles take less time to learn than sudoku — the technique set has a tight upper bound.
What does not scale
Speed-solving has additional tricks that do require more techniques: partial placements where a tile is intentionally left out of position, then corrected at the very end; long sequences of moves planned in advance so the hand can stream them without thinking; pattern-recognition shortcuts that skip the explicit corner manoeuvre for common configurations.
These are not strategically necessary. They make solves faster but they are not part of the minimal technique set.
What you'll discover by yourself
Three things that you should not look up but will figure out:
Tile pairs that are "almost in place" can sometimes be left for the corner manoeuvre. Don't always place tiles in strict numerical order.
The corner manoeuvre direction matters. Top-right corners rotate clockwise; bottom-left rotate counter-clockwise; mirror image. You'll get this right or wrong many times before it's automatic.
Bigger boards reward planning the empty's route in chunks of 3–5 moves, not move-by-move. After about a hundred solves you start moving the empty by intentions rather than individual taps.
What this lets you do
With these four techniques internalised, you can:
- Solve any 3×3 in under 30 seconds.
- Solve any 4×4 in 3–7 minutes.
- Solve any 5×5 in 10–20 minutes.
- Solve any 6×6 in 20–40 minutes.
Those are the time budgets for the row-and-column method. No further strategy lowers them. To go faster, you start cutting corners — but that is a different game and not what this article is about.
Four techniques, every size. That is the strategy.