Rows from the top
Begin at the top, filling the first row tile by tile from left to right. Once it is done, never touch it again: the remaining board is one row shorter and behaves exactly like a smaller puzzle. Keep going row by row until two rows are left.
The last tile of a row
The final tile of each row cannot simply slide in, because the tile before it is in the way. The usual trick is to park the next-to-last tile at the row's end, bring the last tile directly below it, then move the gap round so the pair slides into place together: the first tile moves left and the second follows it up.
It feels like undoing your work for a moment. It is not — the pair drops into place in a few moves.
The last two rows, column by column
When two rows remain, change direction. Take the leftmost column: put its top tile in place, bring the bottom one directly beneath it using the same pair trick, and lock the column. The board becomes two rows that are one column shorter. Repeat until a 2 × 2 corner is left, which you finish by circling the three tiles round the gap.
Numbers and hints
If a picture has many similar tiles — water, sand, sky — turn on the numbers. They do not change anything in the puzzle; they just show each tile's home. The hint button suggests a next tile. On a 3 × 3 board it follows a shortest path; on large boards it gives a useful nudge and, when the board is too complex, highlights a misplaced tile.