TECHNIQUE 1 OF 4
Legal moves: identify the blank's neighbours
Start by separating the moves the board permits from the moves you would like to make. The tile and blank exchange places; no other tile changes during a classic slide.
The rule and its limit
A tile is movable exactly when its square shares a side with the blank. An interior blank has four neighbours, a non-corner edge blank has three, and a corner blank has two. Clicking a distant tile cannot pull it through the intervening tiles.
A common mistake
Treating a diagonal neighbour as adjacent, or imagining that the blank can jump to an empty-looking destination, produces a plan that cannot be played.
PLAYABLE EXAMPLE 1
Find the adjacent tiles
Name every tile that can slide into the blank, then finish the board. A diagonal neighbour does not count.
Loading playable study…
Select a tile beside the blank. Coordinates label every square; rows start at the top.
Keyboard: Tab to an enabled control, then press Enter or Space. The study starts from the diagram shown; reload can restore your own saved moves.
Try the task before inspecting the answer.
Compare the starting moves
The shortest finish takes 1 move. Initially movable tiles: 8, 12, 11. The distance comes from a complete breadth-first search outward from the target through 12 moves; it is not a general shortest-solution claim for arbitrary arcade shuffles.
| First move | Immediate effect | Checked consequence |
|---|---|---|
| Tile 8 (R2C4) | Blank moves to R2C4; Manhattan total becomes 2. | Exact remaining distance: 2 moves. Adds work compared with a shortest finish. |
| Tile 12 (R4C4) | Blank moves to R4C4; Manhattan total becomes 0. | Exact remaining distance: 0 moves. Starts a shortest finish. |
| Tile 11 (R3C3) | Blank moves to R3C3; Manhattan total becomes 2. | Exact remaining distance: 2 moves. Adds work compared with a shortest finish. |
PLAYABLE EXAMPLE 2
Read a central blank
Name every tile that can slide into the blank, then finish the board. A diagonal neighbour does not count.
Loading playable study…
Select a tile beside the blank. Coordinates label every square; rows start at the top.
Keyboard: Tab to an enabled control, then press Enter or Space. The study starts from the diagram shown; reload can restore your own saved moves.
Try the task before inspecting the answer.
Compare the starting moves
The shortest finish takes 2 moves. Initially movable tiles: 7, 15, 10, 11. The distance comes from a complete breadth-first search outward from the target through 12 moves; it is not a general shortest-solution claim for arbitrary arcade shuffles.
| First move | Immediate effect | Checked consequence |
|---|---|---|
| Tile 7 (R2C3) | Blank moves to R2C3; Manhattan total becomes 3. | Exact remaining distance: 3 moves. Adds work compared with a shortest finish. |
| Tile 15 (R4C3) | Blank moves to R4C3; Manhattan total becomes 3. | Exact remaining distance: 3 moves. Adds work compared with a shortest finish. |
| Tile 10 (R3C2) | Blank moves to R3C2; Manhattan total becomes 3. | Exact remaining distance: 3 moves. Adds work compared with a shortest finish. |
| Tile 11 (R3C4) | Blank moves to R3C4; Manhattan total becomes 1. | Exact remaining distance: 1 move. Starts a shortest finish. |
A nearby tile may still be out of reach
On a four-by-four board, the blank is R4C4 and tile 11 is at R3C3. The remaining squares are occupied; their values do not affect this immediate legality question.
The tempting move
Slide tile 11 diagonally into the blank.
Why it fails
R3C3 and R4C4 touch only at a corner. The legal source squares for this blank are R3C4 and R4C3.
What you can conclude
Move a side neighbour first if you need to route the blank toward tile 11. Diagonal proximity does not permit a diagonal slide.
Try the idea in another position
The blank is at R2C2. Which of R1C2, R2C3, R3C3 and R4C2 can supply the next tile? Where is the blank after you move the tile from R2C3?
Check your reasoning
R1C2 and R2C3 share a side with the blank and are legal. R3C3 is diagonal and R4C2 is two rows away. After the R2C3 tile moves into R2C2, the blank is at R2C3.