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Separate the part you control from the new tile
A move has two stages. First, every row or column slides toward the chosen edge, equal neighbours merge once, and the remaining gaps close. Then a changed move places a new 2 in an empty square. The first stage is fully determined by the board; the new tile’s position is not. This lab shows all four deterministic first stages from the same starting board. It does not advance one preview into the next or choose the future spawn for you.
Worked example: four equal tiles
The top row is 2, 2, 2, 2. A left move produces 4, 4, 0, 0 in that row and earns 8 points. It does not make an 8 immediately: the two newly created 4s have already used their merge for this move. On a later move they may combine.
| Direction | Merge points | Empty cells before spawn | Possible spawn squares |
|---|---|---|---|
| Left | 32 | 10 | 10 |
| Right | 32 | 10 | 10 |
| Up | 16 | 7 | 7 |
| Down | 16 | 7 | 7 |
One merge does not necessarily buy more space
Suppose there are E empty cells before a changed move and m pair merges during it. Sliding preserves the number of occupied cells; each merge removes one. Before the fresh tile, there are E + m empty cells. After the single new 2 appears, there are E + m − 1. A pure slide consumes one empty cell. One merge preserves the previous empty-cell count. Two merges gain one. This count does not measure the quality of their positions: a gap trapped behind unlike tiles may be less useful than an open row.
Direction determines the first available pair
A row containing 2, 2, 4, 0 becomes 4, 4, 0, 0 when pushed left. From the right, the original 4 stays in front of the new 4, yielding 0, 0, 4, 4. Neither move combines those two results immediately. Compare where the large tiles land, which rows remain flexible and what the next added 2 could obstruct. The highest immediate score alone is not a proof of the best long-term move.
A no-op is different from a poor move
If every tile is already against the chosen edge and no compatible pair is encountered, the board does not change. The live game adds neither a new tile nor a counted move. A board-changing move that earns zero points still receives a new 2. Treat those as different cases when you count remaining space. If a full board has no legal merge in any direction, the game is over; a single unproductive direction does not establish that.
Try a transfer question
Load “Full board, four winning slides.” Predict which directions can create the 128 target before opening the comparison. There are adjacent 64s at the right end of the last row and in the last column’s lower two cells. All four directions can form 128, but leave it in different locations. This is an immediate finishing example, not evidence that every corner strategy is safe. The tool accepts tiles up to 64 because reaching 128 ends this version of Doubling Tiles.
Updated 4 October 2026. Original calculations and teaching examples; method and limits.