The overlap method
If you learn one nonogram technique, make it this one. The overlap method is how you extract certain filled squares from a clue before you know exactly where its run sits — and it's the engine behind most opening moves. This guide covers the reasoning, a one-line formula, and how it works when a clue has several numbers. It builds on the first-moves guide; if a run and a clue aren't familiar terms yet, start with the rules.
The idea: slide it both ways
Take any single run and imagine pushing it as far to one end of its line as it will go, then as far to the other end. Those are the two extreme positions the run could occupy. Any square that is covered in both positions must be filled — there's no arrangement of that run that leaves it empty. The squares covered in only one of the two extremes stay unknown for now.
That's the whole method. You don't need to know where the run actually goes; you only need the squares it can't avoid.
A one-line formula
For a single run of length k on a line of n squares, the number of guaranteed squares is 2k − n, sitting in the middle of the line — whenever that number is positive. A run of 4 on a 5-wide line gives 2×4 − 5 = 3 forced squares; a run of 3 gives just 2×3 − 5 = 1. The closer the run's length is to the whole line, the more it pins down.
The same idea scales up. A run of 7 on a 10-wide line forces 2×7 − 10 = 4 squares in the middle. You rarely need to compute the number explicitly once you've got the habit — the slide-both-ways picture gives you the same answer by eye.
Single-run overlap
4 on a 5-wide line: 2×4 − 5 = 3 forced squares in the middle.
3 on a 5-wide line: 2×3 − 5 = 1 forced square, the center.
7 on a 10-wide line: 2×7 − 10 = 4 forced squares.
Overlap with more than one clue
When a line's clue has several numbers, the method still works — you just measure the wiggle room shared across the whole line. Add up all the runs plus the single-square gaps required between them; subtract that from the line length. The result is the "slack" — how far the whole arrangement can shift. Each individual run then forces (its own length − the slack) squares, wherever that's positive.
Take a 10-wide line with the clue 4 3. The runs plus one gap need 4 + 1 + 3 = 8 squares, so the slack is 10 − 8 = 2. The run of 4 forces 4 − 2 = 2 squares, and the run of 3 forces 3 − 2 = 1 square. Even a busy line hands you three certain squares before you've placed anything exactly.
Multi-clue overlap: 4 3 on a 10-wide line
Slack = 10 − (4+1+3) = 2. The 4-run forces 2 squares, the 3-run forces 1 — three certain squares in all.
When overlap gives you nothing
If a run is short relative to its line — small k, big n — the formula goes to zero or negative, and overlap forces nothing there. That's completely normal and not a dead end. It just means that line isn't where the opening is; find a more constrained row or column, make progress there, and the squares you fill will shrink the slack on this line until overlap starts to bite. Overlap is rarely the whole solution, but it's almost always the way in.