Line logic
Nonogram solving techniques
Almost every deduction in a nonogram happens inside a single line. These are the four things a line will tell you, in the order it tells you them.
Every deduction happens inside one line
A nonogram looks like a grid problem and is solved as a stack of one-dimensional ones. You take a single row or column, work out everything its own clues force, then move on. What makes the grid a grid is only this: a square settled in a row changes the column crossing it, which is why a line that gave nothing on the first pass often gives something on the third.
Measure the slack before anything else
Add the clue lengths together, then add one empty square between each neighbouring pair. Subtract that total from the length of the line. What is left is the slack — how far the whole arrangement can slide. A 10-wide line with clues 4 3 needs 4 + 1 + 3 = 8 squares, so its slack is 2.
Slack is the number every other technique is measured against. At slack 0 the line is completely determined and you fill it outright. When a clue is longer than the slack, that clue covers squares no matter where it sits. When every clue is shorter than the slack, the line has nothing to give yet and your attention is worth more somewhere else.
The four techniques, in the order a line gives them up
Overlap comes first: any clue longer than the slack fills squares before you know where its run actually is. Edge logic comes next, because once a square at the end of a line is settled the first clue either starts there or is pushed past it, and both outcomes settle more squares.
Forcing follows: a filled square that only one clue can reach pins that clue, which usually collapses the rest of the line. Gap analysis closes it — a stretch of open squares shorter than any clue that could still fit there gets crossed out entirely, and those crosses are what the crossing lines read next.
Sweep rows, then columns, then repeat
Work every row top to bottom, then every column left to right, then start again. Do not chase one interesting line around the grid: the value of a sweep is that each pass leaves the next one more to work with, and a full pass is how you notice it. Most 15×15 grids come apart in three or four sweeps.
If a complete sweep of both directions adds nothing at all, the puzzle has not stalled — your reading of it has. Recount the slack on the two or three longest lines. A miscounted gap is a far more common cause than a genuinely hard deduction.
Common questions
- Which nonogram technique should I learn first?
- Overlap. It is the only technique that fills squares in a line you know nothing else about, so it is what gets an empty grid started. Every other technique needs at least one square already settled.
- Do I ever need to guess in a nonogram?
- No. Every puzzle on this site is verified to have exactly one solution reachable by deduction, so a square you cannot justify is a square that some other line settles first. Guessing is also expensive: a wrong branch stays consistent for a dozen moves before it collapses, and by then you cannot tell which mark was the bad one.
- What does it mean when a line has zero slack?
- It means the clues and their mandatory gaps use the whole line, so there is exactly one way to arrange them. Fill it in completely — a zero-slack line is the cheapest deduction on the board and worth scanning for before anything else.
- How do I solve a 15×15 without losing track?
- Sweep rather than wander. Work every row top to bottom, then every column left to right, then start again. Each pass leaves the next one more to work with, and a full pass is how you notice that a line has become solvable.