The first technique
The overlap technique
The first thing to try on any line: when a clue is longer than the line's slack, it fills squares before you know where its run actually sits.
Push the run both ways
Take a run as far left as it will go, then as far right as it will go. Those are the two extremes of everywhere it could possibly be. Any square the run covers in both extremes is a square it covers in every arrangement in between, because a run is unbroken — it cannot skip over a square it spans at both ends of its travel.
The arithmetic
Slack is the line's length minus the clues and their mandatory gaps. A clue of length c on a line with slack s covers c − s squares in every arrangement. When c − s is zero or less, that clue overlaps nothing and you move on.
Take a 7 on a 10-wide line. It is the only clue, so the slack is 10 − 7 = 3 and the overlap is 7 − 3 = 4 squares. They sit at squares 4 through 7, in the middle of the run's travel. Four squares of a line you otherwise know nothing about, from one subtraction.
Several clues share one slack
Slack belongs to the line, not to any single clue, so the same subtraction runs against each clue separately. On a 15-wide line with clues 4 6, the minimum length is 4 + 1 + 6 = 11 and the slack is 4. The 4 overlaps nothing, because 4 − 4 = 0. The 6 overlaps two squares, because 6 − 4 = 2. One clue paying out while its neighbour pays nothing is the ordinary case, not a sign you have miscounted.
Where overlap is worth spending
Overlap pays best on the longest clue in the tightest line, so scan for lines whose numbers nearly fill them rather than working in grid order on the first pass. On a 15×15 board the opening sweep usually seeds a dozen or so squares this way, and every later deduction is built on those.
When overlap gives nothing
If every clue is shorter than or equal to the slack, the line has no overlap at all. That is not a stall. It means this line needs a crossing square before it can say anything, and crossing squares come from the lines that did pay out — which is the whole reason for sweeping the board in both directions instead of solving one line to the end.
Common questions
- How do I calculate a nonogram overlap?
- Subtract the slack from the clue. Slack is the line's length minus the clues and the mandatory gaps between them; a clue of length c on a line with slack s covers c − s squares in every arrangement. If c − s is zero or less, that clue overlaps nothing.
- Why are the overlapping squares always in the middle of the run?
- Because a run is unbroken. Slide it as far left as it goes and as far right as it goes, and the squares it covers at both extremes are the ones it cannot avoid in between — those sit in the middle of its range of travel.
- Does overlap work when a line has several clues?
- Yes, and each clue is measured separately against the same slack, because slack belongs to the line rather than to one clue. On a 15-wide line with clues 4 6 the slack is 4, so the 4 pays nothing and the 6 pays two squares. One clue paying and the other not is the normal case.
- What do I do when a line has no overlap?
- Leave it. No overlap means the line cannot say anything until a crossing square is settled, and those squares come from the lines that did pay out. Coming back to it after a full sweep costs nothing; staring at it costs the sweep.