Sliding Puzzle

How to play

Tiles sit in a grid with one square empty. Tap any tile next to the gap and it slides in, leaving its own square empty. Work the numbers back into order, reading left to right and top to bottom, with the blank finishing in the bottom right corner. Every shuffle here is checked for solvability before you see it, so no arrangement you are given is impossible. Three sizes are available, and the classic 15 puzzle is the four by four.

Tips

• Solve the top row first, then the left column, then treat what is left as a smaller puzzle and repeat.
• Finish each row with its last two tiles together. Placing them one at a time tends to knock the first one out again.
• Think in terms of moving the blank space, not the tiles. The gap is the only thing you actually control.
• Set the last two tiles of a row by parking them below their final spots, then rotating them in as a pair.
• The bottom two rows must be solved together as a block, because a row of three cannot be fixed in isolation.

The puzzle that fooled everyone in 1880

The 15 puzzle caused a craze in 1880, and a large part of the craze was a version that could not be solved: two tiles swapped, and no amount of sliding would fix it. The reason is a piece of mathematics called parity. Every slide changes the arrangement in a way that preserves a certain property of the permutation, so exactly half of all possible arrangements are reachable and the other half never are. A puzzle dealt from the wrong half is impossible, and looks identical to one that is not.

The game checks parity before every shuffle, so you are never dealt an impossible board. On odd width boards the rule is that the number of out of order pairs must be even; on even width boards it also depends on which row the blank is in.

Solving it methodically

• Solve the top row first, then the left column, then treat what remains as a smaller puzzle.
• Place the last two tiles of a row together. Put them in the row below their targets, then rotate the pair up as a unit.
• Think about moving the blank, not the tiles. The blank is the only thing you control.
• The final 2 by 2 block resolves by cycling. Three tiles and the blank rotate around the block until they line up.
• The bottom two rows have to be finished together, working column by column from the left.

Common questions

Can I be dealt an unsolvable puzzle?

No. Half of all arrangements are impossible, so the game checks parity before every shuffle and only deals solvable boards.

Why are some arrangements impossible?

Every slide preserves a parity property of the arrangement, so from a solved board only half the arrangements can ever be reached.

What is the score?

Moves. Your best for each size is saved separately on this device.