Why Some Sliding Puzzles Can't Be Solved
If you have ever spent an hour on a sliding puzzle and ended with two tiles swapped, it may not have been your fault. Exactly half of all possible arrangements of a sliding tile puzzle can never be solved, no matter how you slide. The famous "14-15 puzzle" from the 1880s, with only 14 and 15 swapped, is one of them.
The reason: every slide keeps a hidden balance
Read the tiles row by row like a list (ignore the empty space). Count the inversions: pairs where a bigger number comes before a smaller one. Sliding a tile left or right never changes this list. Sliding a tile up or down moves it past a few tiles and changes the count in a predictable way. Because of this, some arrangements can never reach the solved order.
How to check a board
- Odd width (3×3, 5×5, 7×7): the board is solvable when the number of inversions is even.
- Even width (4×4, 6×6, 8×8): count the row of the empty space from the bottom (bottom row = 1). The board is solvable when inversions plus that row number is odd.
Why the 14-15 puzzle fails
In the solved 4×4, there are 0 inversions and the empty space is on row 1 from the bottom: 0 + 1 = 1, odd, solvable. Swap 14 and 15 and you add exactly one inversion: 1 + 1 = 2, even, impossible.
What this means for you
A physical puzzle that was taken apart and put back carelessly may be unsolvable. A good puzzle app checks this rule on every shuffle. In Tesslide every board is guaranteed solvable, and if you get lost, a hint shows a move that leads to the finish.
FAQ
Can I fix an unsolvable board?
Only by swapping two tiles, which you cannot do by sliding. In a physical puzzle that means popping two tiles out. In an app, start a new shuffle.
Are Tesslide puzzles always solvable?
Yes. Every shuffle is checked with the parity rule, and hints can show a next move from any position.
Is a 3×3 different from a 4×4?
The rule is slightly different for odd and even widths, but in both cases exactly half of all arrangements are solvable.