DECISION STUDY · slide-07
Check the odd invariant
Calculate inversions while ignoring the blank. Add the blank’s row counted from the bottom. Is this position reachable from the target? Then solve it.
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.
Show the starting-position answer and evidence
The shortest finish takes 6 moves. There are 9 inversions and the blank is in row 4 from the bottom. Their sum 13 is odd, matching the target’s parity. Swapping two numbered tiles while keeping the blank fixed would make this sum even and that altered board impossible. 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 1 (R2C1) | Blank moves to R2C1; Manhattan total becomes 5. | Exact remaining distance: 5 moves. Starts a shortest finish. |
| Tile 2 (R1C2) | Blank moves to R1C2; Manhattan total becomes 7. | Exact remaining distance: 7 moves. Adds work compared with a shortest finish. |
One complete checked route
These moves start from the original diagram, not from any different choices you made above. Read coordinates at the moment each move is made.
- Slide Tile 1 (R2C1) into the blank at R1C1.
- Slide Tile 5 (R2C2) into the blank at R2C1.
- Slide Tile 6 (R3C2) into the blank at R2C2.
- Slide Tile 10 (R3C3) into the blank at R3C2.
- Slide Tile 11 (R3C4) into the blank at R3C3.
- Slide Tile 12 (R4C4) into the blank at R3C4.
After the first move
Verified finish
How this position was checked
A fixed reachable position selected from legal moves outward from the target.
Explain what you learned
Choose one first move from the comparison table. Explain its effect without rereading the answer. Then change the first move and identify which part of the explanation changes. A checked route answers this starting position; it does not automatically apply to a new one.
Rules and limits
This version uses a four-by-four grid containing tiles 1–15 and one blank. Only a tile directly above, below, left or right of the blank can slide into it. The goal is 1–15 in reading order with the blank at R4C4. There are no diagonal slides, edge wrapping or whole-row slides. The game creates its scrambles through legal moves from the goal.
A legal move need not improve a position. Local routes and cycles demonstrate their stated effects; they do not prove a globally shortest solution. A study's minimum-distance claim requires a search that has ruled out every shorter route from that exact position. It is not a promise about general scrambles. The parity test below applies to this four-by-four board and this particular goal. A route that preserves a solved region is useful only while enough space remains to execute it. Generated scrambles are reachable, but an arbitrary rearrangement supplied in a study may not be.