Solvability guide
Are Slide Puzzles Solvable?
Yes, many slide puzzles are solvable, but not every arrangement is. Whether a board can be solved depends on parity: the order of the numbered tiles, and on even-width boards, the row of the empty square.
The short answer
A slide puzzle is solvable only if its tile order matches the parity of the solved board. Legal moves preserve that parity, so an impossible board cannot be fixed by trying harder. It is not a matter of skill or move count; the board is mathematically unreachable.
This matters most when a puzzle is created by randomly placing tiles. If the board is scrambled by making legal moves from the solved position, it will always be solvable. If tiles are placed randomly, roughly half of the possible arrangements will be unsolvable.
What is an inversion?
An inversion is a pair of numbered tiles that appear in the wrong order when you read the board from left to right, top to bottom, ignoring the empty square.
Example sequence:
Tile 8 appears before tile 7. Since 8 is larger than 7, that pair is one inversion.
The 3x3 slide puzzle rule
- 1 Write the tile numbers in reading order, skipping the empty square.
- 2 Count every pair where a larger number comes before a smaller number.
- 3 If the total is even, the 3x3 puzzle is solvable. If the total is odd, it is not solvable.
This is why a 3x3 board like 1 2 3 / 4 5 6 / 7 empty 8 has zero inversions and is solvable, while the board with only 7 and 8 swapped has one inversion and cannot be solved.
The 4x4 slide puzzle rule
A 4x4 puzzle, often called the 15 puzzle, uses the inversion count plus the empty square row. Count the empty row from the bottom, not from the top.
Solvability rules by puzzle size
| Puzzle | Board width | Solvability rule |
|---|---|---|
| 3x3 / 8 puzzle | Odd | Solvable when the inversion count is even. |
| 4x4 / 15 puzzle | Even | Solvable when the empty row from the bottom and inversion count have opposite parity. |
| 5x5 and other odd-width puzzles | Odd | Same idea as 3x3: solvable when the inversion count is even. |
| 6x6 and other even-width puzzles | Even | Same idea as 4x4: include the empty row from the bottom. |
Why can a puzzle look almost solved but still be impossible?
The classic impossible position is a solved board with the last two numbered tiles swapped. It looks close because every tile except those two is in place. But a legal slide puzzle move cannot swap just two tiles while leaving the rest of the board unchanged.
Legal moves change the board in cycles. Those cycles preserve the parity rule. That is why the impossible board stays impossible even if you move tiles around for hundreds of turns.
How to make sure a slide puzzle is solvable
- Start from the solved board and scramble it with legal moves.
- Use a solvability test before showing a randomly generated board.
- For a 3x3 board, count inversions and require an even total.
- For a 4x4 board, count inversions and the empty row from the bottom.
FreeSlidePuzzles.com uses generated board patterns that are validated as solvable before they are included in the game.
Common slide puzzle solvability questions
Can every 3x3 slide puzzle be solved?
No. Only 3x3 boards with an even inversion count are solvable.
Can every 4x4 slide puzzle be solved?
No. A 4x4 board must pass both the inversion count rule and the empty-row rule.
Why are randomly shuffled slide puzzles sometimes impossible?
Random placement ignores parity. About half of random arrangements land in the unreachable half of the puzzle state space.
Is a puzzle solvable if it was scrambled by legal moves?
Yes. If the board started solved and was scrambled only by legal slide moves, then reversing those moves will solve it.
Can the solver find the shortest solution?
The 3x3 solver checks solvability and returns an optimal solution with the fewest moves for that board.
Check your 3x3 board
Arrange the tiles exactly as they appear on your puzzle. The solver will tell you whether the board is solvable and, if it is, show the optimal solution.