The fourth technique
Gap analysis
Once a line has a cross in it, it is no longer one line but two. Measure each piece against the clues that could still live there, and the pieces too short to hold any of them are empty end to end.
A cross cuts a line into segments
A crossed-out square is a wall. Nothing passes through it, so a line carrying one cross is really two shorter lines, and each of them has to hold whichever clues can still reach it. That is the whole technique: stop looking at the line and start looking at its pieces.
Take a 12-square line with a single clue of 5. Its slack is 7, so overlap finds nothing at all. Now cross out square 4. The piece to its left is three squares long and a run of five cannot fit in three, so squares 1 to 3 are empty as well — and the piece to the right is eight squares holding a 5, which overlaps squares 8 and 9. One cross produced three more crosses and two filled squares.
Closing one segment can close the line
When the surviving segment is exactly as long as the clues it must hold, the line has no slack left anywhere and it resolves completely. This is the most valuable thing gap analysis does, and it happens more often than it sounds.
A 15-square line with clues 4 6 gives you squares 10 and 11 from overlap and no more. Cross out square 4 and squares 1 to 3 go with it, because neither clue fits in three squares. That leaves squares 5 to 15 — eleven squares for 4 + 1 + 6 = 11. There is exactly one arrangement left: the 4 on squares 5 to 8, square 9 empty, the 6 on squares 10 to 15. The whole line, from one cross.
The segment you must not cross out
The rule is that a segment is empty when no clue that could still reach it fits — not when the biggest clue does not fit. Checking only the largest number is the way this technique goes wrong, and it goes wrong quietly, because a wrongly crossed square stays consistent for a long time before the grid contradicts it.
Change the clues to 2 6 and cross square 4 again. The three squares on the left still cannot hold the 6, but they hold the 2 comfortably, so nothing there can be crossed. The cross was not worthless — it shortens the space the 6 has to live in, which fills square 10 — but the three squares you were about to claim are not yours.
Mark the crosses, not just the fills
Gap analysis only works on a board where crosses are actually written down, because the segments it measures are the ones the crosses create. A player who marks filled squares and leaves proved-empty squares blank keeps the information in their head, where the crossing lines cannot read it, and then wonders why a grid that should sweep clean has stalled.
Common questions
- What is gap analysis in a nonogram?
- Measuring the stretches of open squares a line has been cut into, and crossing out any stretch too short to hold a clue that could still reach it. It is the technique that turns one cross into several, which is why it usually runs last on a line rather than first.
- How do I know which clues can still reach a segment?
- A clue can only reach a segment if every clue before it still fits in the space to the left, and every clue after it still fits to the right. If no clue passes that test for a given segment, the whole segment is empty.
- Why are crosses worth as much as filled squares?
- Because the crossing lines read them. A cross tells the line running through it that a run stops there, which is what turns a long open line into two short measurable ones. Players who only mark filled squares lose most of the information a solved line produces.
- When does gap analysis run in a sweep?
- After the line has picked up marks from overlap, an edge, or a forcing square. It needs at least one cross or a completed run to have something to measure against, and it pays best immediately after a run has been closed off.