TECHNIQUE 2 OF 4
Blank routing: prepare the destination first
To move a wanted tile in a particular direction, put the blank on that side of it. This often requires moving other tiles before touching the tile you are trying to place.
The rule and its limit
Name the tile's intended next square, then find a legal path for the blank to that square without moving the wanted tile prematurely. Every step of the blank's path is an exchange with another tile. A tile becoming farther from its goal during this preparation is not, by itself, proof that the plan is wrong.
A common mistake
Clicking the wanted tile whenever it becomes movable may send it in the wrong direction. Moving only tiles that immediately approach their final squares can also prevent a needed repositioning of the blank.
PLAYABLE EXAMPLE 1
Make space before moving a tile
Plan the blank’s route before chasing a numbered tile. Compare the first moves, then find a shortest finish.
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 5 moves. A tile can travel only into the current blank. Use the complete route below to see which setup moves create that space. 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 6. | Exact remaining distance: 6 moves. Adds work compared with a shortest finish. |
| Tile 15 (R4C4) | Blank moves to R4C4; Manhattan total becomes 6. | Exact remaining distance: 6 moves. Adds work compared with a shortest finish. |
| Tile 12 (R3C3) | Blank moves to R3C3; Manhattan total becomes 4. | Exact remaining distance: 4 moves. Starts a shortest finish. |
PLAYABLE EXAMPLE 2
Route around displaced tiles
Plan the blank’s route before chasing a numbered tile. Compare the first moves, then find a shortest finish.
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 7 moves. A tile can travel only into the current blank. Use the complete route below to see which setup moves create that space. 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 9 (R2C2) | Blank moves to R2C2; Manhattan total becomes 6. | Exact remaining distance: 6 moves. Starts a shortest finish. |
| Tile 10 (R4C2) | Blank moves to R4C2; Manhattan total becomes 6. | Exact remaining distance: 8 moves. Adds work compared with a shortest finish. |
| Tile 5 (R3C1) | Blank moves to R3C1; Manhattan total becomes 8. | Exact remaining distance: 8 moves. Adds work compared with a shortest finish. |
| Tile 11 (R3C3) | Blank moves to R3C3; Manhattan total becomes 8. | Exact remaining distance: 8 moves. Adds work compared with a shortest finish. |
A legal click can move the wanted tile the wrong way
The blank is R1C3. A wanted tile at R2C3 should next move left to R2C2. All squares on the proposed route lie within the four-by-four board.
The tempting move
Select the wanted tile immediately because it is adjacent to the blank.
Why it fails
That click moves the tile up to R1C3, not left to R2C2.
What you can conclude
To make the intended left move, first select R1C2 and then R2C2, routing the blank around the wanted tile. Now select R2C3. The sequence moves the wanted tile into R2C2, although the displaced surrounding tiles still need consideration.
Try the idea in another position
The blank is R2C2 and a wanted tile is at R3C3. You want that tile to move up into R2C3. Which two source squares can you select, in order, to do this?
Check your reasoning
First select the tile at R2C3, moving it left into the blank. The blank is now R2C3. Next select the wanted tile at R3C3, moving it up. The blank finishes at R3C3. This describes the local effect, not a claim that the whole board is solved.