Why a sliding puzzle can be unsolvable

An unsolvable sliding puzzle has no sequence of legal slides that reaches the selected target. This is a property of the board and its goal, not a sign that a search merely needs more time. The sliding puzzle solver checks this relationship before searching. A valid input can be impossible; an input with a duplicate or missing tile is instead an entry error.

Check the copy before changing anything

Compare every entered row with the original board, including the blank. Make sure you selected the correct size and target. The usual target is ascending numbers with the gap at bottom-right, but a game can use another layout. Use More → Edit goal when needed.

For picture pieces, verify their identities against the completed reference. Two adjacent pieces of sky or a repetitive background can look interchangeable but stand for different numbered cells. Uploading an image does not perform that recognition. The picture mapping guide explains how to enter the correspondence accurately.

Odd-width boards: 3x3, 5x5 and 7x7

For ascending numbers and the blank at bottom-right, write the numbered tiles in row order and ignore the blank. Count inversions: pairs where a larger tile comes before a smaller one. An even count is solvable; an odd count is not.

For example, 1 2 3 / 4 5 6 / 8 7 0 has one inversion, the pair 8 and 7. That is an impossible 3x3 arrangement for the usual goal. Moving the blank around cannot turn it into the solved board. The blank's row is not part of this particular odd-width rule.

Even-width boards: 4x4, 6x6 and 8x8

With the same ascending target and bottom-right blank, add the inversion count to the blank's row counted from the bottom. The bottom row is 1, the row above it is 2, and so on. An odd sum is solvable; an even sum is not. Counting from the top by mistake changes the test.

The solved 4x4 board has zero inversions and a blank in bottom row 1. Its sum is 1, which is odd. The row term matters because vertical moves on an even-width board change both the numbered sequence's parity and the blank row's parity.

Worked parity check: tiles 14 and 15 swapped

 1  2  3  4
 5  6  7  8
 9 10 11 12
13 15 14  0

There is exactly one inversion: 15 comes before 14. The blank is in row 1 from the bottom. Add them: 1 + 1 = 2. The sum is even, so this board cannot reach the normal 15-puzzle goal through slides.

Load this exact impossible board to reproduce the warning. No Solve operation is required to establish that the parity is wrong. Contrast it with the solvable three-move example: that board has three inversions and a blank in row 2 from the bottom, giving the odd sum 5.

What Fix it does

The repair button exchanges two neighboring numbered tiles in the virtual arrangement. Exchanging one pair changes the parity class, making the modified board solvable for the current target. The blank is not one of the exchanged tiles. Undo restores the original entered arrangement.

This exchange is not a legal sliding move. If the original board is a physical toy or a game that permits only slides, the repaired sequence cannot be applied to the unchanged original. First correct an input mistake if there is one. If the physical board really has the impossible arrangement, it needs a physical reassembly or another action allowed by that game before a sliding solution exists.

Custom goals and the general rule

The quick inversion formulas above assume a particular goal. For a custom goal, label each tile by its position in that goal, include the blank, and compare the resulting permutation parity with the parity of the blank's Manhattan distance between start and goal. They must agree. The solver performs this goal-relative check, so you do not need to convert a custom target into the standard numbering rule yourself.

If there is one blank but tiles have irregular shapes, several cells are blocked, or the board is rectangular, the tool's square-board rules do not apply. For a supported board that passes the check, continue with how to follow a computed solution or the 15-puzzle walkthrough.

Mathematical reference

The parity distinction and the difference between odd and even board widths are described in the University of Alberta sliding-tile teaching notes. The worked arrangements on this page were also checked with this site's goal-relative solvability function. They describe the selected standard target, not every possible target.