PUZZLES
Sliding Puzzle Guide: Solve Any Board Row by Row
The classic way to solve a sliding puzzle is to complete one row at a time, then finish the last two rows together. The method works on boards of any size and needs no memorised sequences.
The idea behind the method
You cannot solve the whole board at once, so you build it from one corner and never disturb what is already finished. Each completed row or column makes the remaining puzzle smaller, until only a tiny corner is left. The only difficult moment is placing the last tile of a row, and there is a simple trick for that.
Solve the top row first
Start with the first tile, then the second, and so on along the row. For the last tile in the row, slide it into the square directly below its target, bring the previous tile into the corner above it, and then rotate the pair into place with the blank. The trick is to set the last two tiles up together rather than placing them one at a time.
Then the left column
After the first row is done, treat the remaining board as a smaller puzzle. Solve the left column of that smaller puzzle using the same technique, turned ninety degrees. Alternate rows and columns until a two-by-two area remains.
Finish the last two by two
With three tiles and the blank in a two-by-two area, the tiles can only rotate around the blank. If they are in the right cyclic order, a few rotations finish the puzzle. If they are not, an earlier step needs to be redone, and that is a good time to use undo.
Why some boards cannot be solved
Exactly half of all arrangements of a sliding puzzle can be solved. A solvability test counts inversions, the number of pairs where a larger tile precedes a smaller one, and combines that with the blank square’s row on boards of even width. PlayDapple only deals boards that pass this test, so every puzzle you see has a solution.
Efficiency tips
Think in cycles. Moving the blank around a two-by-three block rotates the tiles in a loop, which can reposition a tile without disturbing those already placed. Plan the whole loop before you start, and use undo if you lose track.
Larger boards mostly take longer, not harder, because the method is the same. The first rows of a 5×5 board go quickly, and the real work is in the last few.
A worked example on a 3×3 board
Suppose the top row is already solved with tiles one, two and three, and you are working on the left column, tiles four and seven. Place tile four directly under tile one, then move tile seven to the position immediately to the right of the target, move the blank above it, and slide seven down. The last tile of a column is placed using the same set-up trick as the last tile of a row, turned ninety degrees.
After that, only the bottom-right two-by-two area remains. Rotate the last tiles until they are in order. The whole process takes a few dozen moves on a 3×3 board and gets longer, but not harder, on larger boards.
Common mistakes
The most common mistake is placing the last tile of a row directly, which pushes the tile you already placed out of position. Always set up the last two tiles together. Another is trying to fix one stubborn tile by disturbing the solved rows, which usually undoes more than it gains.
Moves and efficiency
Solving by the row method is not the shortest path, but it is reliable. Expert players use their knowledge of which tiles are close to their positions to take shortcuts, but even they use the row method as a fallback. If you want to reduce your move count, look for moves that place two tiles at once.