Line Solving: One Row or Column at a Time

A line's clues describe that line completely, independent of the rest of the grid: the numbers are the runs of filled squares, in order, each separated by at least one empty square. So before you ever look at how a row interacts with its columns, you can ask a narrower question — given only these clues and this length, which squares must be filled and which must be empty no matter how the runs end up positioned?

The answer is often "some of them," and finding those forced squares one line at a time is the whole engine of solving. You rarely finish a line in a single pass, but you almost always learn something — and that something becomes a clue for every line that crosses it.

Here is the general method that covers every line, however many clues it has. Mentally slide all the runs as far to the left (or top) as they will go, respecting their order and the one-square gaps between them. Then do the same as far to the right (or bottom). Any square that is covered by the same run in both extreme packings can never be empty — it is forced filled.

Take a 10-square line with the clues 3 4. Packed hard left, the runs sit at squares 1-3 and 5-8. Packed hard right, they sit at 3-5 and 7-10. Compare the two: the first run covers square 3 both times, and the second run covers squares 7 and 8 both times. So squares 3, 7 and 8 are guaranteed filled, and you can mark them immediately — even though you don't yet know exactly where either run begins.

This is exactly the overlap trick, just generalized: with a single long clue it reduces to the familiar "slide it to both ends" move, but the pack-left-pack-right framing keeps working when a line has two, three or more runs, which is where eyeballing the overlap starts to fail.

Line solving gets much stronger once a line already contains some information from its crossings. A square you've marked empty (an ✕) acts as a wall: it splits the line into shorter segments, and each run must fit entirely within one segment — which often pins runs that were free a moment ago. A square you've marked filled anchors whichever run must cover it, so you can extend that run and cap its ends.

That is why solving is a loop, not a single sweep. Work a row as far as it goes, and its new marks — filled squares and ✕s alike — travel down into every column that crosses it. Work those columns, and their new marks travel back into the rows. Each line becomes solvable a little further every time one of its crossings gains information.

A line is "done for now" when its current clues and known squares force nothing more — not necessarily when it's complete. Don't get stuck grinding one stubborn line; mark everything it currently forces, then move to the line that has gained the most new information since you last looked at it. On a well-made puzzle, cycling through rows and columns this way keeps unlocking fresh deductions until the whole picture is filled.

If you cycle all the way around and truly nothing moves anywhere, you haven't hit a puzzle that needs a guess — you've missed a deduction on some line. A properly constructed nonogram has exactly one solution reachable by pure logic, so the forced square is always there; line solving is simply the discipline of finding it.