Is every 15 puzzle solvable?
No. Exactly half of all arrangements are solvable. The 15 puzzle has 16! = 20,922,789,888,000 possible arrangements, and 10,461,394,944,000 of them can be solved. A position is solvable when the number of inversions plus the row of the empty square, counted from the bottom starting at 1, is an odd number (the solved board itself scores 0 + 1 = 1). A single sliding move never changes that parity, which is why the other half can never be reached no matter how long you play.
How do you solve the 15 puzzle?
Solve it in layers, never in one pass. Place the top row first (tiles 1, 2, 3, 4), then the left column (5, 9, 13), and treat what is left as a smaller 3x3 puzzle. Repeat until only a 2x2 corner remains, which is solved by rotating the last three tiles. The rule that makes it work is simple: once a row or column is finished, never move a tile out of it again. The last two tiles of each row are placed together using a short setup sequence, not one at a time.
What is the maximum number of moves needed to solve a 15 puzzle?
80 moves. The hardest possible starting position on a 4x4 board takes 80 single-tile moves to solve optimally, a result proven by exhaustive computer search. This number is often called God's number for the 15 puzzle. If moving a whole row or column at once counts as one move, the figure drops to 43.
Do I need to install anything to play?
No. Slide15 runs in the browser, free, with no signup and no download. There is also a native iPhone and iPad version with hand-crafted shape levels and a daily challenge, but nothing on this page requires it.
Can I play the 15 puzzle with a keyboard?
Yes. Use the arrow keys to slide the tile next to the empty square. On desktop you can also click or drag a tile, and on a phone or tablet you can swipe. All three input methods work on the same board.
What is the difference between the 15 puzzle and the 8 puzzle?
Board size, and nothing else. The 15 puzzle is 4x4 with 15 numbered tiles and one gap; the 8 puzzle is 3x3 with 8 tiles. The solving method and the parity rule are identical, but the 8 puzzle has only 181,440 solvable positions against the 15 puzzle's 10.5 trillion, which is why it is used for teaching and the 15 puzzle for playing.