Stop chasing tiles. Start thinking in loops.
Replay a four-move loop on a fixed board, reverse it, and see exactly which tiles stay in place.
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. Load a board you can check
In Tile Shift, choose Load guide example beneath the live board. It replaces the current puzzle with the arrangement below and sets the move counter to zero. Rows count from top to bottom, columns from left to right, and the dot is the empty square. You can use a paper copy instead if you prefer.
1 2 3 4 5 6 7 8 9 10 12 15 13 14 11 ·
The upper two rows and the left half of the lower two rows are already correct. The working area is only rows 3–4, columns 3–4: tiles 12, 15, 11, and the gap. This board is made by four legal slides from the solved board, so the example is reachable.
Follow one four-move loop
Select the numbered tiles 11, 12, 15, then 11 again. Each selection moves one tile into the adjacent gap. The small diagrams below show only the working area; the other twelve cells stay unchanged.
Move 1: select 11 at row 4, column 3. It slides right. The gap is now at row 4, column 3.
12 15 · 11
Move 2: select 12 at row 3, column 3. It slides down. The gap is now at row 3, column 3.
· 15 12 11
Move 3: select 15 at row 3, column 4. It slides left. The gap is now at row 3, column 4.
15 · 12 11
Move 4: select 11 at row 4, column 4. It slides up. The gap returns to the bottom-right corner.
15 11 12 ·
The counter should read 4. The gap is back where it started, but the numbered tiles have rotated: returning the gap does not restore the board. Tile 11 was selected twice from different positions. Follow tile numbers, not a memorized sequence of fixed button positions.
3. Reverse the loop, then try a solution
From that four-move result, select 11, 15, 12, then 11. These are the earlier tile selections in reverse order. Every tile and the gap return to the full starting diagram, and the counter reads 8. Reversing a slide is still a legal move, so it still counts.
Choose Retry this board to restore the same starting diagram with zero moves. This also works if you wandered away from the exercise or finished the puzzle. New puzzle is different: it creates a random arrangement. Loading the example again also resets it; neither action saves the puzzle you replace.
For a complete solution of this starting example, select 15, 12, 11, then 15. The working area becomes 11, 12 above 15 and the gap. All sixteen cells now match the goal, and the game reports a solution in four moves. This solution goes around the same small loop in the other direction. These numbers solve this fixed example only; a random board needs its own plan.
Why is returning the gap not the same as undoing the loop?
A circuit moves three numbered tiles into one another’s former places. The gap returns after four slides, while the tiles remain rotated. Undoing the circuit requires reversing the actual slides, as in the eight-move round trip above.
4. 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.
The worked loop demonstrates controlled local movement, not a universal sequence for every last pair. 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.
5. Practice a controlled experiment
On a random board, choose a two-by-two area containing the empty square. Write down the three tile numbers and predict where each will finish after a loop. Compare your result with that prediction. If you want a fresh attempt, Retry this board restores the arrangement created by the latest New puzzle or either example-loading action; it does not rewind just one move. Refreshing a fixed practice entry reloads its example with zero moves; refreshing a random-mode page creates a new random puzzle.
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.