Nonograms, Explained

Read this and you will understand how a nonogram turns a picture into number clues, why some pictures make good puzzles and others do not, and how to check whether your puzzle can be solved by logic alone.

What a nonogram is

A nonogram is a grid puzzle. You start with a blank grid, and the clues along each row and column tell you where the filled cells go. When you finish, the filled cells form a picture. The puzzle is also sold as picross, griddler and hanjie.

Each clue is a list of numbers. A row clue of 3 1 means: reading left to right, there is a run of 3 filled cells, then at least one gap, then a run of 1 filled cell. The numbers never tell you where the runs start. You deduce that from the other clues.

Take a tiny 5x5 grid holding a plus sign. The center column is fully filled and the center row is fully filled. The column clues, left to right, read 1, 1, 5, 1, 1. The row clues, top to bottom, read the same: 1, 1, 5, 1, 1. From those ten short lists you can rebuild the plus with no guessing. That is the whole idea: a picture compressed into run lengths, and a solver who expands it again.

When to use it, and when not

Nonograms suit a printed worksheet, a puzzle booklet, or an embed on a lesson page. They teach reasoning about constraints, and they reward patience rather than speed. A single 15x15 puzzle occupies most people for 15 to 40 minutes.

They are a poor fit when you want a quick warm-up (a 5x5 takes under two minutes and feels slight) or when the solver has no pencil-and-paper habit, because marking known blanks is half the technique. They also fail when the picture is a poor source. A grid that is nearly empty or nearly full carries little information, and the clues stop pinning down a single answer.

Counting the possibilities in one line

To see why density matters, count how many ways a single row can be filled. Suppose a row has n cells and a clue with k runs whose lengths sum to s. The runs need k-1 mandatory gaps between them, so the fixed cells total s + (k-1). The free space you can distribute is n - s - (k-1) cells, spread across k+1 slots (before, between, after the runs).

Here N is the number of legal placements for that row, n is the row length, s is the total of the clue numbers, and k is how many numbers the clue has. The binomial \binom{a}{b} counts how many ways to choose b gap positions from a options.

Check it on the plus row 5 in a width-5 grid: n = 5, s = 5, k = 1, so N = \binom{1}{1} = 1. One placement. Now a clue of 1 in the same row: n = 5, s = 1, k = 1, giving N = \binom{5}{1} = 5. Five placements. That single filled cell could sit anywhere, and only the column clues resolve it.

A fully filled or fully empty line has N = 1. A half-filled line has the most placements. That peak is exactly where ambiguity hides, which is why balanced pictures with clear edges solve cleanly and noisy half-densities do not.

A worked example: the demo puzzle

How to read and interpret the clues

Two techniques cover most deductions. The first is overlap. If a row has width 10 and a single run of 8, the run can start at cell 1, 2 or 3. In all three positions cells 3 through 8 are filled, so you can fill those six cells immediately. The overlap length is s - (n - s) when there is one run, which here is 8 - 2 = 6.

The second is edge forcing. If cell 1 is already known blank and the leftmost clue is a run of 3, that run cannot touch the left wall, so it slides right. Combined with column deductions, forced blanks cascade. Marking blanks with a small dot or X is not optional: a blank you have proved is as useful as a fill you have proved.

Common mistakes when building puzzles

The most frequent error is drawing a picture that reads clearly to a human but leaves the grid ambiguous. A face made of four isolated dots (two eyes, two nostrils) gives four rows of 1 and several columns of 1, and those single cells can swap positions without changing any clue. Connect features or add a border to lock them down.

The second error is ignoring the empty-line warning. An empty row contributes clue 0, which tells the solver nothing and often lets neighbouring runs float. A puzzle with three or more empty rows in a 10x10 grid is worth re-checking for a second solution.

The third is over-dense art. A silhouette that fills 70 percent of a 15x15 grid produces long clues that overlap heavily, so the puzzle solves in two minutes and feels flat. Aim for 30 to 45 percent fill with recognizable connected shapes. That range keeps enough lines at low N to give footholds while leaving real work to do.

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If you like grid logic that guarantees a single answer, the Sudoku Generator builds puzzles with a unique solution at five difficulty levels. Sudoku and nonograms share the same discipline: deduce, never guess, and mark what you have ruled out.

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