TECHNIQUE 3 OF 4
Small cycles: return the blank while moving the tiles
A loop can restore the blank to its starting square while changing the nearby tile order. Trace the contents as well as the blank so you know what the loop actually accomplishes.
The rule and its limit
Consider a two-by-two region with blank at its top left, tile A at top right, B at bottom left and C at bottom right. Select top right, bottom right, bottom left and top left. The blank returns to top left, while C moves to top right, A to bottom left and B to bottom right. Squares outside the region are untouched.
A common mistake
The blank returning to its start does not imply that every tile returned too. Conversely, walking the blank out and immediately retracing the same path can simply undo the work.
PLAYABLE EXAMPLE 1
A loop changes tile order
Watch the displaced tiles as you route the blank. Reach the target, then compare your route with the checked shortest route.
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 4 moves. Moves 1–4 in the checked route circle a two-by-two block: the blank returns to R4C4 while three tiles rotate. Moving a tile out and immediately back is a two-move undo. A four-step trip around a 2×2 block instead cycles three tiles; it is not generally a reset. 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 15 (R3C4) | Blank moves to R3C4; Manhattan total becomes 3. | Exact remaining distance: 3 moves. Starts a shortest finish. |
| Tile 11 (R4C3) | Blank moves to R4C3; Manhattan total becomes 5. | Exact remaining distance: 5 moves. Adds work compared with a shortest finish. |
PLAYABLE EXAMPLE 2
Compare a longer cycle
Watch the displaced tiles as you route the blank. Reach the target, then compare your route with the checked shortest route.
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 8 moves. Moves 2–5 in the checked route circle a two-by-two block: the blank returns to R3C2 while three tiles rotate. Moving a tile out and immediately back is a two-move undo. A four-step trip around a 2×2 block instead cycles three tiles; it is not generally a reset. 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 2 (R1C2) | Blank moves to R1C2; Manhattan total becomes 9. | Exact remaining distance: 9 moves. Adds work compared with a shortest finish. |
| Tile 6 (R3C2) | Blank moves to R3C2; Manhattan total becomes 7. | Exact remaining distance: 7 moves. Starts a shortest finish. |
| Tile 5 (R2C1) | Blank moves to R2C1; Manhattan total becomes 9. | Exact remaining distance: 9 moves. Adds work compared with a shortest finish. |
| Tile 7 (R2C3) | Blank moves to R2C3; Manhattan total becomes 9. | Exact remaining distance: 9 moves. Adds work compared with a shortest finish. |
Going out and back is not a productive cycle
The blank is at R2C2 and tile A is at R2C3. Select R2C3, then select R2C2.
The tempting move
Count the two-move sequence as a useful rearrangement because the blank has moved around a tile.
Why it fails
The first move puts A at R2C2 and the blank at R2C3. The second swaps them straight back. No tile has changed its final position.
What you can conclude
This pair cancels as a rearrangement, although it still consumes two moves. Use a loop with a verified tile effect when you need to reposition several tiles.
Try the idea in another position
After the four-move loop just described, the region reads blank/C on the top row and A/B on the bottom. What happens if you repeat the same four source-square selections twice more?
Check your reasoning
Each loop cycles the three tiles. Three loops in total return A, B and C to their original squares, with the blank still at top left. This explains the loop's effect; repeatedly cycling without a target does not make progress.