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Puzzle Quarry

Nonogram Solving Techniques

One question asked five times, with a little more of the line uncovered at each attempt. Where a board stops being answerable is the level it ships with.

One question, asked with less of the line hidden each time

Every rung on this ladder asks the identical question: given these numbers, what do all the legal arrangements of this line agree on? The only thing that changes from one rung to the next is how much of the line you are allowed to look at while you ask it. That is not a tidy way of describing five separate tricks — it is how the solver behind this site is built, and it is why the order is not a matter of taste.

The first rung looks at none of the line and reads only the numbers beside it. The second reads the two ends: whatever is already settled at the start and at the finish, and nothing in between. The third adds the crosses, which is what cuts the line into pockets. The fourth adds the filled blocks you can see at a glance. The fifth takes the line in as it is, one or two stray filled cells included.

Because each rung reads strictly more than the one below, a board that needs the fourth genuinely could not be finished with the third — and that, rather than a count of numbers or a size, is what the level on every board here means.

The label is a measurement, not an estimate

Every published board is run through the same solver the hint button uses. It applies the cheapest argument that fits, over and over, until every cell is decided, and it records the highest rung it ever stood on. Easy stops at the ends of the lines. Medium reaches the crosses. Hard reaches the blocks. Expert reaches a line that has to be taken in whole.

Nothing is graded by how many numbers a board carries and nothing is relabelled afterwards. The generated practice boards routinely come out at a level other than the one requested, and when they do they ship under the level measured. The hand-drawn boards cannot be steered at all — a drawing is what it is — so their labels are simply what the solver found.

There is a sixth rung and nothing here is allowed to stand on it. Filling a cell on spec, following it until a line breaks and rubbing it out again will finish any board whatsoever, which is exactly why it cannot be part of a difficulty scale. Its page explains how to do it properly if you meet one elsewhere, and why every board needing it is thrown away here.

The two middle rungs came out the other way round

Attaching a filled block to a run feels like the easier of the two middle arguments, and the ladder was first written with it below the one about crosses. It is the wrong way round, and the reason is not a matter of opinion: you cannot attach a block to anything until you know where the walls are, so finding the walls has to come first. A rung that reads the crosses is a smaller look than one that reads the crosses and the blocks against them.

The measurement settled it. Run several thousand generated boards through both orderings and with the block argument placed lower, the fourth rung is the hardest step required on about one board in fifty — a difficulty level with nothing in it. Swapped round, all four levels carry real weight at every size.

The same measurement decided what counts as a block you can see. A run of three or more filled cells is an object whether or not anything is beside it; one or two adrift in an open stretch are not, and using them is what the top rung is for. Set that line anywhere else and either the top rung swallows a third of every board or it empties completely.

Common questions

How many techniques does a nonogram need?

Five, and most boards need three of them. The numbers alone and the ends of the lines will carry an easy board; the three above them are what each further level of difficulty means here.

Which nonogram technique should I learn next?

Whichever one your current level stops at. Stalling on medium means reading the crosses as walls is not yet a habit; stalling on expert means working a whole line out is not. The hint names the argument it used.

More Nonogram pages