Tile Shift: protect a solved row while the other tiles circulate
Use a four-move loop to understand how tiles circulate, then distinguish a useful boundary from restrictions that leave too little room to work.
A completed row is a useful boundary
After placing 1, 2 and 3 across the top of a three-by-three board, it is tempting to declare them untouchable. Keeping the blank in the lower two rows reduces the area you need to watch. Instead of reconsidering every tile after each slide, you can concentrate on the six remaining positions.
Keep that row as a clear working boundary, but be careful about imposing the same restriction on every correctly placed tile below it. The remaining tiles need room to circulate. Insisting that each lower tile stay fixed as soon as it looks right can make the available routes unnecessarily restrictive.
Watch a small loop change three tiles
Begin with the completed board: 1 2 3 / 4 5 6 / 7 8 blank. Follow the blank through four directions: Left, Up, Right, Down. These directions move the empty square around the bottom-right two-by-two block. The top row, tile 4 and tile 7 remain untouched throughout the loop.
The resulting board is 1 2 3 / 4 6 8 / 7 5 blank. The blank has returned to its original corner, but tiles 5, 6 and 8 have each changed position. Returning the blank to its starting square therefore does not necessarily restore the board. A loop can circulate tiles while preserving its surroundings.
Reverse a sequence in the correct order
To undo that four-move loop, take its directions in reverse order and reverse each direction. Down becomes Up, Right becomes Left, Up becomes Down, and Left becomes Right. The return sequence is therefore Up, Left, Down, Right. Applied to the changed board, it restores the completed position.
This is a useful way to record a short experiment. Write the blank’s directions, then work out the inverse before moving. Simply swapping Left with Right in the original order is not enough: the blank would visit different squares, so the intended slides might no longer be legal.
Protect positions that are not part of the current route
Before using a loop elsewhere, identify every square it crosses. Those are the tiles you are allowing to change. Squares outside the route stay fixed. This makes a loop more predictable than an unplanned collection of slides, but it does not make any chosen loop the right move for your current puzzle.
Compare the board before and after the full sequence. Did a target tile get closer to the position you need, and can the blank reach the next useful side of it? Looking only at the first slide can be misleading, because a productive circulation may initially move a tile away from its destination.
Reconsider restrictions inside the unfinished region
If your chosen small loop recreates a position you have already seen, pause rather than repeating it faster. Consider the whole unfinished two-row region. A route through more of those squares may temporarily move a correctly placed lower tile while letting another pass. Record its return path before you begin the experiment.
The example here explains circulation and reversal; it is not a universal sequence for finishing the last two rows. Your board determines which tiles need to circulate. Use Undo to return from a failed experiment, and preserve the reasoning you gained even when you decide to restart the puzzle.
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