2048

Choose one of four directions. Every tile on the board slides that way until it hits a wall or another tile, and any two tiles carrying the same number that come together in the slide fuse into a single tile showing their sum — two 8s become a 16, and you score 16 points for it. As soon as a move has changed something, one new tile appears in a randomly chosen free cell. Keep fusing until a 2048 appears.

The whole game is that loop, and the difficulty is entirely in the tile that arrives afterwards. You control four directions; the board answers with a tile you did not choose, in a cell you did not choose. Every decision is really about where you are willing to let that tile land.

• Swipe anywhere on the board, or use the arrow keys, W A S D, or the direction pad under it.
• A move that changes nothing is not a move — no tile appears, no move is counted, and the undo stack is untouched. Pressing into a wall is free.
• Undo walks back up to 24 moves. Hint searches the position and lights up a direction.
• The game ends when the board is full and no two neighbouring tiles match. A full board on its own is not a loss.

Because a tile may take part in only one merge per move, and merges resolve from the direction you are travelling. Slide that row left and the pair nearest the left wall fuses first; the 4 it produces is finished for the move, so it cannot swallow the 4 arriving behind it. You get 4, 4 — never 8. This is the rule that separates a correct implementation from a broken one, and it only shows itself on a full line of equal tiles, which is why so many versions get it wrong.

Slide that same 4 4 2 2 row right instead and it still scores 12, but the board ends up different: the 2s are the pair nearest the right wall now, so they fuse first and settle against it, leaving · · 8 4 rather than 8 4 · ·. One rule, read from whichever wall you are pushing towards.

On the standard board, 131,072. The argument is short. Suppose the grid is full and the largest tile is 2^m. That tile was built from two half-sized tiles, one of which had to be parked somewhere while the other was assembled, and the same is true all the way down the ladder — so in the very best case the cells hold 2^m, then 2^(m-1), then 2^(m-2), down to the smallest tile the game ever hands you, which is a 4. That is m − 1 distinct tiles in n² cells, so m = n² + 1.

The consequence for the small board is worth stating plainly: a 3 × 3 board cannot produce a 2048 tile at all. Nine cells put the ceiling at 1,024. But a ceiling is not a target — hitting 1,024 on nine cells would need those cells to hold 1024, 512, 256, 128, 64, 32, 16, 8 and 4 simultaneously, with nowhere for the tile that arrives next, and the search behind the hint button never once got there in the twenty-four small boards it is set to play, finishing none of them above 512. So the 3 × 3 board asks for 512, which the same search finished on in five games out of twenty-four. Setting the small board to 2048 — or to its own ceiling — would be setting a target that cannot be met, and some implementations do exactly that.

Longer than it looks, and the reason is arithmetic rather than skill. Merging never changes the sum of the numbers on the board — an 8 and an 8 leave as a 16 — so the total showing at any moment is exactly the total of every tile that has ever been handed to you. A 2048 tile therefore needs 2,048 points of spawned tiles, and the most generous spawn is a 4. That is 512 spawns; two of them are the opening tiles and every other one costs a move, so 510 moves is the floor, and it assumes every single tile that ever appeared was a 4.

In reality nine tiles in ten are 2s, so a real game runs roughly twice that. Pinned to its shallowest two-ply setting and left to play whole games by itself, the search behind the hint button reached 2048 on at least four of sixteen seeded 4 × 4 boards and never finished below 512 — and the successful runs needed between 1,005 and 1,206 moves. At a move a second — about as fast as anyone can actually read a board — that is seventeen to twenty minutes for a 4 × 4 win, and rather less for a 5 × 5, where nine extra free cells make the same target much gentler.

The score arithmetic is just as fixed. A merge pays the face value of the tile it creates, so building a tile of value v entirely out of 2s costs v × (log₂ v − 1) points: a 2048 made purely from 2s is worth exactly 20,480. Every 4 that spawns saves you one merge and four points, which is why a real 2048 usually shows up a little under that figure, with the rest of the board making up the difference.

One time in ten, and it is worth being careful about which tiles you count. Over the 40,000 opening tiles of 20,000 seeded games, 10.0% came up 4; over all 175,138 spawns that 1,500 complete games actually consume — the ones you meet in the middle of a board, not just the two you start with — it is 10.0% again. Because the two opening tiles are drawn independently, about one game in a hundred opens with two 4s: 548 of 50,000 seeded openings here, or 1.1%.

A 4 looks like a gift and usually is not. It carries twice the value of a 2, but it cannot merge with the 2s around it, so it sits in the middle of your ladder as a rung with nothing above or below it until you have built a second 4 to meet it. A run of early 4s on a 3 × 3 board is close to fatal. On 5 × 5 you will barely notice.

Yes. Every board is a pure function of the 32-bit number printed in the status strip — both opening tiles, and every tile that appears afterwards. Game #482137 deals the identical sequence on a phone, a laptop and inside the app, today and in five years, and there is no server anywhere in that: the sequence is derived from the seed and the number of tiles already spawned. That is also what lets a half-finished game be saved into the browser and picked up later playing exactly the same run of tiles it would have played.

Different numbers are genuinely different games, which is easier to get wrong than it sounds — derive the tile stream carelessly and game #482136 turns out to be game #482137 with its tiles shuffled between neighbouring turns. Here the first 30,000 game numbers open on 30,000 different sequences, and two numbers one apart are no more alike than two numbers a million apart.

Replay this game on the end-of-game card and you get the same opening and the same run of tiles behind it — play the same moves and the board is identical cell for cell, which is the only honest way to find out whether a game was lost to the board or to you. Undo is the smaller version of the same idea: it walks back up to 24 positions, and because the seed and the spawn count travel with the position, redoing the move you took back deals the tile it was always going to deal.