Stop chasing tiles. Start thinking in loops.
Use the empty square as a tool, and move a group of tiles without undoing the whole board.
The most tempting move in a sliding puzzle is to push a number toward its final square. That can work for a while, but it misses half of the problem: the empty square has to be in the right place before the tile can move. A better starting question is, “Where does the empty square need to be next?”
1. Separate the tile from the route
If tile 6 needs to travel left, the empty square must be immediately to its left. You may need several moves to put the empty square there. Those moves can look unproductive because tile 6 does not move at all. They are preparation, much like moving a chair before opening a door.
Choose one target tile and one target square. Trace the route the empty square can take without disturbing the portion of the puzzle you have already completed. If the route crosses a finished row, check whether a loop below that row could bring the gap around another way.
2. See what a four-square loop does
Consider this local two-by-two section. The dot represents the empty square. Select C, then A, then B, then C again. The gap returns to its starting corner, while the three numbered tiles rotate.
Start After C After A After B After C A B A B · B B · B C C · · C A C A C A ·
The rest of the board does not change. This is why loops are useful: you can rearrange a small group without dragging the gap across everything you have solved. Notice that C is selected twice, from different positions. The sequence describes tiles, not fixed screen coordinates.
3. Finish a row without trapping its last pair
Getting 1 and 2 into the top row is usually straightforward. Finishing 3 and 4 can be less forgiving. If 3 is fixed too early, tile 4 may need to pass through the square occupied by 3. Temporarily move the pair into a small working area, line them up in the order they need, and rotate them into the row together.
Do not interpret “finish a row” as “never move any correct tile.” A temporary step backward can open the only useful path. What matters is whether your short sequence restores the earlier work at its end.
4. Practice a controlled experiment
On a fresh board, choose a two-by-two area containing the empty square. Make one full loop and observe which tiles changed places. Then reverse your exact sequence. You should return to the same board. This exercise teaches you to predict the effect of a move sequence before committing to a larger rearrangement.
What this method does—and does not—solve
Loops are a building block, not a complete shortest-path algorithm. The last two rows need more care because there is less working space. Our starting positions are generated by legal slides from the solved board, so you do not need to diagnose whether a shuffled board is impossible. Focus on controlled movement first; efficient solutions can come later.