Sliding Puzzle

How to play

The tiles are numbered from 1 upwards and one square is left empty. A tile can only move into that empty square, and only if it is directly beside it, so at any moment you have at most four legal moves and often only two or three. The board is finished when the numbers read left to right, top to bottom, with the gap sitting in the bottom right corner. Tiles that are already in their final position are tinted green, which saves you counting.

Nothing here is random in the dangerous sense. A common way to build one of these puzzles is to throw the tiles into random positions, and it produces an unwinnable board roughly half the time — a genuinely infuriating experience, because the board looks completely normal right up until two tiles refuse to swap at the end. This one starts from the solved state and plays several hundred random legal slides before handing it over. A sequence of moves can always be walked backwards, so the position in front of you always has a way home.

Three sizes are offered. The 3 × 3 board is a warm-up you can talk yourself through. The 4 × 4 is the classic fifteen-tile puzzle and the one the method below is written for. The 5 × 5 uses the same technique with more rings to peel and roughly triple the patience. Your choice of size is remembered between visits, along with a separate best move count and best time for each.

Controls

• Click or tap a tile that touches the gap and it slides across. Tapping any other tile does nothing at all, which is the honest response.
• Arrow keys or W A S D move the gap. Press up and the gap travels up, which means the tile that was sitting above it comes down.
• Swipe on a touch screen describes the tile rather than the gap: swipe right and the tile to the left of the gap travels right into it.
• Target hints attaches each tile's home row and column to its hover text and to the label a screen reader reads out. It is a lookup aid, not a solver — no arrows appear and nothing moves on its own.
• Shuffle deals a new arrangement at the current size and resets both counters. The clock starts on your first move, not on page load.

Solve it in rings, not in numbers

The instinct is to chase tile 1, then tile 2, then tile 3, and to be pleased with each one. The trouble arrives around tile 7, when you realise that every route to the tile you want runs through the tiles you already placed. Fixing anything breaks everything, and the board starts to feel like a machine designed to punish progress.

The method that fixes this is to shrink the board instead of filling it. Complete the entire top row, then the entire left column. Once both are done, treat them as walls: never move the gap into them again. What is left of a 4 × 4 is a 3 × 3 puzzle, which obeys exactly the same rules. Solve its top row and its left column, and you are down to a 2 × 2 corner, which is three tiles rotating around a single gap and can be finished by spinning them until they land.

The working method inside a ring is small. To bring a tile somewhere, you move the gap, not the tile: walk the gap around to the far side of the tile you want to shift, then push the tile one square by moving the gap back through it. Everything in a sliding puzzle is really a sequence of gap journeys, and thinking of it that way stops you from planning moves you cannot make.

The two-tile problem at the end of every row

Rows do not finish cleanly, and this is the part that stops most people. Suppose you are completing the top row of a 4 × 4 and tiles 1 and 2 are already home. Place 3 in its final square and then try to bring 4 in from below: there is no way to slot 4 into the corner without pushing 3 out of the way again. You can loop that forever.

The way out is to build the pair somewhere else and insert them together. Put tile 3 into the corner square where 4 belongs, and park tile 4 directly beneath it. The two of them are now stacked in the wrong place but in the right relationship. Bring the gap round to the left of tile 3, slide 3 left into its home, then bring 4 up into the corner. Two tiles land at once and nothing you already placed is disturbed.

The same trick, rotated, closes every column: stack the last two tiles of the column side by side in the wrong place, then rotate them in. Once the pattern is familiar the 4 × 4 stops being a puzzle and becomes a procedure, and the interesting question shifts from whether you can finish to how few moves it takes you.

Where the puzzle came from

The fifteen-tile version set off a genuine craze in 1880, in the United States and then across Europe, helped along by a prize offered for solving a position that could not be solved. Noyes Palmer Chapman, a postmaster in New York State, is generally credited with the original design; the puzzle seller Sam Loyd later claimed the invention as his own, and historians have never been able to support that claim.

The impossible prize position is the reason the puzzle has a place in mathematics. In 1879 mathematicians published the argument showing that the arrangements split into two families of equal size, and that no sequence of slides can ever carry you from one family to the other. Exactly half of all the ways of laying out the tiles are reachable from the solved board, and the other half are not, which is why the shuffle on this page is built from real moves rather than a random deal. The implementation is our own, written for this site.

Questions players ask