How to solve nonograms: reading the numbers correctly

A nonogram (also called picross or a griddler) hands you a grid and, for every row and every column, a short list of numbers instead of a picture. Reading those numbers correctly is the whole game, once you know what they mean, filling in the grid becomes a logic puzzle rather than a guessing exercise.

What the numbers actually mean

Each number in a row or column’s clue is the length of one unbroken run of filled cells somewhere along that line, and the numbers are listed in the order those runs appear. A clue of “3 2” means: somewhere in this line there’s a run of exactly 3 filled cells, then later, a run of exactly 2 filled cells, with the 3-run appearing before the 2-run. Between separate runs in the same clue there has to be at least one empty cell, otherwise they’d merge into a single longer run and the clue would list them as one number instead of two. There can be more than one empty cell between runs, the clue only guarantees a minimum gap of one, not an exact gap.

The overlap technique

You don’t need to know exactly where a run starts to know that some of its cells must be filled. If a single run is more than half the length of its line, sliding it as far left as possible and as far right as possible still leaves some cells covered by the run in both positions, and those cells are guaranteed filled no matter where the run actually ends up.

Take a line of 10 cells with a single clue of 7. Pushed as far left as it can go, the run covers cells 1 through 7. Pushed as far right as it can go, it covers cells 4 through 10:

cells:      A B C D E F G H I J
leftmost:   # # # # # # # . . .
rightmost:  . . . # # # # # # #
overlap:          # # # #

Cells D, E, F, and G (positions 4 through 7) are filled in both the leftmost and rightmost placements, so they’re filled in every placement in between too, the run can only slide, not shrink. Those four cells can be marked filled immediately, before you’ve committed to where the run actually starts. Cells A–C and H–J stay unknown for now, since the run could cover some of them or none of them depending on exactly where it lands.

The general rule: for a single run of length n in a line of length L, the number of guaranteed-filled cells works out to 2n - L, and only counts for anything when that’s a positive number, which happens whenever a run is more than half the line’s length (in the example above, 2(7) - 10 = 4, the four overlap cells found earlier). A run of 5 in a line of 10 has no guaranteed overlap at all (it can slide entirely clear of any single fixed cell), which is why this technique only pays off on the longer runs.

Working with more than one run

When a line has multiple runs in its clue, the same idea still applies, but each run has to leave room for the others. The leftmost arrangement packs every run as far left as possible with exactly one empty cell between each pair; the rightmost arrangement does the mirror image, packed to the right. Comparing the two placements run by run gives you the overlap for each one individually. It’s more bookkeeping than the single-run case, but the underlying logic, filling in whatever’s covered in both the leftmost and rightmost placement, is exactly the same.

After the overlaps run out

Overlap fills in the easy, guaranteed cells, but it won’t solve a whole puzzle by itself. Once a line stops yielding new overlap cells, the next step is usually checking rows and columns against each other: a cell you’ve filled in from a row’s clue also narrows down what that cell’s column can look like, and vice versa. Working back and forth between rows and columns, re-checking each line as its neighbors fill in, is how the rest of the picture gets resolved.