Nonograms: the harder line techniques

Everything happens one line at a time

It is worth saying clearly, because it changes how the puzzle feels: a nonogram is never solved by looking at the whole grid. Every single deduction you will ever make comes from one row or one column, considered on its own, using its clue and whatever is already marked in it.

The grid only matters as a courier. A square proved in a row is delivered to a column, which then has more information than it did, which produces another square. Progress spreads like a crack rather than sweeping like a scan.

So the practical skill is squeezing a single line completely dry before moving on. The techniques below are how you do that. If you have not read the basics, the overlap method there is the prerequisite for all of this.

A blank splits a line into shorter lines

This is the most under-used technique in the puzzle. A square you have proved empty does not just remove one possibility — it cuts the line into independent segments, and a run can only live inside a segment long enough to hold it.

Do this check every time you mark a blank. Work out the segments the line now has, and for each remaining run ask which segments could possibly hold it. Two outcomes are common and both are valuable:

• A run fits in only one segment. Apply overlap inside that segment alone, which is much tighter than overlap across the whole line.
• A segment is too short for every remaining run. The whole segment is blank. Mark it and split again.

Pinning a run against an edge

Edges are the most informative part of any line, because a run touching one has nowhere to slide. A single filled square at the very start of a line pins the first run completely.

The same logic applies inward. A filled square in the second position means the first run either starts at square one or square two, which is often enough to fix part of it by overlap. And a filled square close to an edge, with a long first run, can be pinned even without touching the edge: if the run cannot reach back far enough to start at square one, its position is more constrained than the raw overlap suggested.

Always work both ends of a line. Solvers tend to read left to right and miss the identical deduction sitting at the far end.

Forcing from a single filled square

A filled square in the middle of a line seems to say very little. It says more than it looks, because it must belong to some run, and you can test each run in turn.

For each run in the clue, ask: could this run cover that square? Often only one or two can, because the others would have to overlap a known blank or run off the end of the line. If exactly one run can cover it, you know which run it is, and that pins how far the run can extend in each direction.

Even when two runs remain possible, you frequently learn something. If both candidate runs are at least three long, then the squares immediately adjacent are filled either way — the run has to extend somewhere, and both options cover those squares.

Counting a line to its finish

Lines are frequently finished long before they look finished, and leaving them unresolved starves the crossing lines of information.

Two counting checks are worth running on every line you touch:

1. Add the clue up. If the filled squares already present equal that total, every unmarked square in the line is blank. Mark them all.
2. Count the runs, not just the squares. If all the runs in the clue are accounted for as complete, separated blocks, the line is finished even if you have not consciously decided so.

The second check catches a common stall: a line whose runs are all correctly placed, but whose remaining squares were never crossed off, so the columns crossing it never received the blanks they needed.

Which line to attack next

With a partly-solved grid there are thirty candidate lines and most of them will give you nothing. Choosing well is most of the speed.

• The line you just changed, crossed. After filling a square, read its column. This is where the next deduction is, the great majority of the time.
• Lines with the least slack. Add the clue plus the mandatory gaps and compare to the length. The smaller the difference, the more forced the line.
• Lines that just gained a blank. A new blank means new segments, which means the splitting technique has fresh material.
• Lines with a filled square near an edge. Cheap to check and often decisive.

What not to do is sweep the grid in order. On a fifteen-wide puzzle that wastes most of your attention on lines that have not changed since the last time you read them.

Every picture in Pixel Logic is drawn first and its clues derived from the finished image, so the clues always describe exactly one drawing and it is always reachable by these methods. If a line seems to admit two answers, the deciding information is in the crossing lines.