How to solve nonograms: the overlap technique and other tips

To solve a nonogram, work one row or column at a time and only fill the squares that must be filled whatever happens. The main tool is the overlap technique: slide each run of squares as far left and as far right as it can go, and any square it covers in both positions is certain. Repeat across every line, cross out squares that must be empty, and the picture appears without a single guess.

What a nonogram is

A nonogram — also sold as Picross, Griddlers, Hanjie or paint by numbers — is a grid of squares with numbers along the top and the left side. Each set of numbers describes one line: how many squares in a row are filled, in order, with at least one empty square between each run. Fill in every square the clues describe and a small picture is revealed.

A clue of 3 1 on a row means: somewhere in this row there is a run of three filled squares, then at least one gap, then a single filled square. It does not tell you where they are. Working that out, line by line, is the puzzle.

How our nonograms work

• Daily: one ten-by-ten picture a day, the same for everyone.
• Free play: three sizes — five by five for a quick warm-up, ten by ten, and fifteen by fifteen for a long sit-down.
• Never a guess: every picture has been checked so that it can be finished one line at a time, by logic alone.
• Relaxed by default: a square in the wrong place is just a square in the wrong place — nothing is taken away. Turn on Strict if you would rather a wrong square simply refused to go in.
• Helpful clues: a clue dims once its line adds up, so you can see at a glance what is left.

Press and drag to fill a run in one go; the drag keeps to the row or column you start along. You can switch to crossing out, or right-click on a mouse. On a keyboard, Space fills, X crosses out and H asks for a hint.

Technique 1: the overlap

This is the technique that starts almost every nonogram. Take one line and its clue. Push every run as far left as it will go, then as far right. Any square that is covered by the same run in both positions must be filled, because that run has to cover it wherever it ends up.

Here is a worked example on a ten-square row with the clue 3 4. The clue needs 3 + 1 + 4 = 8 squares at minimum, leaving 2 squares of slack.

Clue: 3 4        Row length: 10
Minimum needed:  3 + 1 + 4 = 8    Slack: 10 - 8 = 2

             1  2  3  4  5  6  7  8  9  10
Leftmost:    A  A  A  .  B  B  B  B  .  .
Rightmost:   .  .  A  A  A  .  B  B  B  B
Certain:     .  .  #  .  .  .  #  #  .  .

A = the run of 3     B = the run of 4     # = must be filled

Run A covers square 3 in both positions, and run B covers squares 7 and 8 in both. Those three squares are certain. Notice square 5: it is filled in both positions, but by different runs — B on the left, A on the right — so it proves nothing. Checking that it is the same run is the step people most often skip.

There is a handy shortcut. For each run, subtract the slack from its length. If the answer is positive, that many squares are certain. Here, 3 − 2 = 1 square for run A and 4 − 2 = 2 squares for run B, which matches the diagram exactly. A clue of 7 in ten squares has a slack of 3, so the middle 7 − 3 = 4 squares are filled.

Technique 2: start with the big clues

The overlap technique works best when a clue fills most of its line. So look first for:

• Full lines. A clue of 10 in a ten-square line fills the whole thing.
• Clues that add up exactly. A clue like 4 5 in ten squares needs 4 + 1 + 5 = 10 — no slack at all, so the line is completely determined.
• Zeros. A clue of 0 means the whole line is empty. Cross it out straight away.
• Large single runs. Anything more than half the line length gives you certain squares in the middle.

Technique 3: use the edges

A filled square touching the edge of the grid is powerful. If the first square of a row is filled, the first run in that row must start there, so you can fill its full length and cross out the square straight after it.

For example, in a row with clue 4 2, if square 1 is already filled from a column, then squares 1 to 4 are the run of 4 and square 5 must be empty. The same idea works from the right-hand edge with the last run in the clue.

Technique 4: cross out what cannot be filled

Crossing squares out is as important as filling them. A cross means “definitely empty”, and every cross you place shrinks the space the runs can move in, which makes the overlap technique work again. Cross out:

• every remaining square once a line’s clue is complete;
• the squares on either side of a finished run;
• squares that are out of reach — too far from any run that could cover them.

On BrainBreak, crosses are only working notes: they are never scored, and the picture is finished when the right squares are filled, crosses or no crosses. But they make the solve far easier to read.

Technique 5: go back and forth

No single line solves a nonogram. The rhythm is: work every row you can, then every column, then back to the rows. Each new filled square or cross in a column is new information for the rows that cross it. When you seem stuck, sweep through every line once more with fresh eyes — there is nearly always a square you can now place.

Common mistakes

• Forgetting the gaps. Runs in the same line need at least one empty square between them. Leave it out of your arithmetic and the overlap comes out wrong.
• Mixing up runs. As in the example, a square being filled in both extremes only counts if it is the same run.
• Guessing. Every one of our puzzles can be solved without it. If you feel you have to guess, look at the crossing lines again instead.
• Not crossing out. Unmarked empty squares look just like unsolved ones, and it is easy to lose your place.

Where to go from here

Start with a five-by-five puzzle to get a feel for the overlap, then move up to the daily ten-by-ten. If you enjoy the careful, one-step- at-a-time logic, you will probably like Sudoku as well — our Sudoku tips for beginners cover the equivalent techniques — and our list of logic games that make you think has more ideas.