The third technique
Forcing
Ask which clue a filled square could possibly belong to. When only one can reach it, that run is pinned, and the rest of the line usually follows.
Every clue has a reach
Pack the runs as far left as they go and you have the earliest square each clue can occupy. Pack them as far right and you have the latest. Between those two positions is that clue's reach, and it can never touch a square outside it. Reaches overlap near the middle of a line and separate towards the ends.
So when you meet a filled square, ask which reaches contain it. If the answer is one, that square belongs to that run, and you have learned something no amount of staring at the clue list would give you.
A pinned run drags the rest of the line with it
Take a 15-wide line with clues 1 1 6. Its slack is 5, so overlap forces a single square — square 10 — and nothing else. Now fill in square 15 from a crossing line. Neither 1 can reach that far, so the 6 owns it, and the only way a run of six covers square 15 is by occupying squares 10 to 15 exactly.
The run is now pinned, which also settles what is around it: square 9 has to be empty, because a filled square there would make the run seven long. One square from a crossing line turned into five more filled squares and a cross.
Check that the deduction is actually new
Forcing is the technique that most often feels productive while producing nothing. On a 12-wide line with clues 2 5, square 8 is already forced by overlap alone. Fill in square 6 and it looks as though the 5 has been pinned — but the only square you can now claim is the one overlap had already given you.
The test is quick: before marking anything, ask whether this line would have given you the same square with nothing known. If it would, you have spent attention rather than earned ground, and the line still needs a crossing square before it will say anything new.
Where forcing pays best
Look for filled squares near the ends of long lines with several clues. That is where the reaches have separated, so a single clue owns the square outright, and it is also where the pinned run is long enough for the deduction to be worth having. Filled squares in the middle of a line with generous slack are usually reachable by two clues at once, and those are the ones to leave alone.
Common questions
- What is forcing in a nonogram?
- Working out which clue a filled square belongs to. Each clue can only reach a limited stretch of the line, so a filled square outside every other clue's reach must belong to the one clue that reaches it — and knowing which run it is constrains where that run starts.
- How do I know how far a clue can reach?
- Pack every run as far left as it goes to find the earliest square each one can occupy, then pack them as far right for the latest. A clue reaches everything between those two extremes and nothing outside them. A filled square inside exactly one clue's reach is a forcing square.
- What if two clues can both reach the square?
- Then forcing tells you nothing yet, and the honest move is to leave the line. Check whether the square you were about to claim was already forced without it — a deduction that only re-proves what overlap gave you is a deduction that cost you the sweep.
- Does forcing ever prove a square is empty?
- Often, yes. Once a run is pinned and its full length is accounted for, the squares immediately before and after it cannot be filled, so they get crossed out. Those crosses are usually worth more to the crossing lines than the filled squares are.