How to play Tile Order

Tiles sit in a square frame with one square left empty. A tile next to the gap can slide into it. Nothing else is legal, nothing is ever added or removed, and the puzzle is finished when the tiles read in order with the gap in the bottom-right corner.

That is the whole game and it has been the whole game since the 1870s. What differs between versions of it is not the rules but whether the game can tell you anything about how well you played.

The number on screen

Most sliding puzzles count your moves and stop there, which tells you nothing: forty moves might be superb or might be twice what the board needed.

On the three-by-three and four-by-four boards this one searches the puzzle before you see it and shows the smallest number of moves that can possibly solve it. The word used is **Fewest**, and it is meant literally — the search is exhaustive in the sense that matters, so no shorter route exists. Finishing on that number is a perfect run and cannot be bettered.

On the five-by-five and six-by-six the same claim would be a lie. Proving a five-by-five optimal is out of reach of a browser and out of reach of most things that are not browsers. Those boards run the game's own solver instead and show what it managed, under the word **Par**. Par is beatable, and beating it is a genuine result rather than a rounding error.

The two are deliberately not called the same thing and do not look the same, because they are different claims. A proven minimum is marked in green; a par is not.

What proving it costs

The proof is affordable only because the boards are built to a known depth.

A fully shuffled four-by-four takes up to twenty-two seconds to prove optimal, which is not something to do while somebody waits. Built by a walk of forty-five moves, the same proof lands in nine milliseconds on a typical board and a fifth of a second on the worst one measured. So the walk is capped at forty-five, and that cap is what buys the number.

The honest consequence is that these four-by-fours are shorter than a completely random one: about thirty-four moves at the top of the ladder rather than the fifty-odd a free shuffle averages. That is the trade, and it is a trade — a shuffle nobody can measure would have been easier to ship.

The three-by-three has no such problem and runs to a full shuffle, where the hardest board possible needs thirty-one moves and a typical one twenty-two.

Every board can be solved

Half of all arrangements of a sliding puzzle cannot be solved at all, and a dead board looks exactly like a live one. You can lose an hour to a puzzle that never had an answer.

The usual fix is to shuffle freely and then repair the board using a parity rule. It works, but it means the same fact — whether this board can be finished — is written down in two places, and two places drift.

This game never shuffles. It starts from the finished board and walks the gap away from it, so every board it produces is solvable because a route home was just walked to build it. There is no parity rule in the game at all, because there is nothing for one to fix. The walk is also kept, and reversed it solves the board, which is where the target comes from on the boards that cannot be proved.

Pushing a run

Tapping a tile that is not next to the gap, but is somewhere along its row or column, slides every tile between the two at once.

This is not a shortcut so much as the thing that makes a six-by-six playable with a thumb. A six-by-six needs a few hundred moves and doing that one square at a time is not a puzzle, it is data entry.

The count still ticks once per tile, because that is the unit the target is measured in and a target measured in a different currency to your score is worthless. Undo, though, takes back the whole push, since that is the thing you actually did.

Reading a board

Two habits do most of the work.

The first is to solve the top row and the left column and then forget they exist. What is left is a smaller puzzle of exactly the same kind, and the whole method is to keep shrinking it until three squares by three remain, which can be finished by feel.

The second is to stop chasing the tile you want. Almost all of the work is moving the **gap** to where it needs to be, and the tile only travels on the one move in four or five where it is in the way. Players who think in terms of driving the gap around improve faster than players who think in terms of dragging tiles.

The corner of a row is the part worth learning properly. The last two tiles of any row cannot be placed one after the other — putting the last one home pushes the other one out. They go in together, by parking the second-to-last tile in the corner and the last one directly beneath it, then rotating both in with one turn.

The ladder

Boards run from three-by-three up to six-by-six, and the shuffle deepens as you clear levels.

It stops at six on purpose. A seven-by-seven takes around six hundred and forty moves and uses not one technique that a five-by-five does not, so it is longer rather than harder, and length is not difficulty.

The daily board

One four-by-four a day, the same for everybody, built from the date — the size that gets a proven minimum, so everyone is chasing the same real number.

Finishing extends your streak, your best move count is kept, and the share text carries your moves, that board's fewest and your time without giving the arrangement away.