Strategy
Nonogram Strategy: 8 Solving Techniques, from Overlap to Contradiction
Every hard nonogram is built from a handful of patterns. Learn to spot these eight and big puzzles stop being intimidating.
The short version: almost every nonogram deduction comes from imagining a run pushed as far left and as far right as it can go, and keeping whatever is true in both positions. The other techniques (reach, edge forcing, joining, capping, contradiction) are refinements of that one idea, applied once some cells are already known.
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This guide assumes you know the basic rules. If not, start with How to Solve Nonograms, which walks through a full puzzle. Each technique below shows a single 10-cell row, because every nonogram is solved one line at a time.
Key for the diagrams: green and gold show where runs could sit, orange cells and orange crosses are what the technique proves.
1. The overlap method (simple boxes)
Take a row of 10 cells with a single clue of 7. Push the run all the way left, then all the way right. Cells 4 to 7 are covered either way, so they must be filled, even though you don't know where the run starts.
The quick formula for a single clue: if the clue k is more than half the line length n, you get 2k − n filled cells in the centre. A 7 in 10 gives 4; a 6 in 10 gives 2; a 5 in 10 gives none.
2. Count the slack on multi-number clues
Overlap works with several clues too. Add the clue numbers plus one cell for each required gap: for 3 4 that's 3 + 1 + 4 = 8. In a 10-cell row there are 2 cells of slack, the amount any run can slide. Every run longer than the slack gives you (length − slack) filled cells.
One trap: only count overlap between the same run. In the diagram, the right-pushed 3 covers cell 5, which the left-pushed 4 also covers, but that tells you nothing, because they're different runs.
Slack is also the fastest way to size up a puzzle. Scan every clue for the lines with the least slack; those are where you start.
3. Reach: cross out what a run can't touch
Once a line has a filled cell, that cell limits where its run can be. Suppose the clue is 2 and cell 5 is already filled (from a crossing column). A run of two that includes cell 5 can only cover cells 4–5 or 5–6. Everything else is out of reach.
4. Edge forcing
Filled cells near a wall are powerful, because the wall stops a run from sliding. With the clue 4 and cell 2 filled, the run must include cell 2, so it starts at cell 1 or cell 2. Either way it covers cells 2, 3 and 4, and it can't reach past cell 5.
The general rule: if the first filled cell in a line is at position p and belongs to the first run of length k, then cells p through k are filled whenever p ≤ k. The same works from the right-hand wall, and from any X you've placed, since an X acts like a wall.
5. Joining separated cells
Two filled cells that must belong to the same run get joined. With the clue 5 and cells 3 and 6 filled, there is only one run, so it must include both, and everything between them fills in. The run then starts at cell 2 or 3, so cell 1 and cells 8–10 are empty.
The reverse is useful too (splitting): if joining two cells would make a run longer than any clue in the line, the cell between them must be empty.
6. Cap finished runs
When a run of filled cells reaches the full length of its clue, put an X on both ends. Then see what that unlocks. With the clue 3 2 and cells 5–7 filled, that run of three must be the 3 (the other clue is only 2). Cap it, clear everything before it, and the 2 has exactly two cells left.
This chain (identify a run, cap it, re-measure the space that's left) is how most of the “hard” middle part of a big puzzle gets solved.
7. Cross-check rows against columns
None of the techniques above uses more than one line, but every cell sits in two lines. The real rhythm of solving is:
- Apply every technique you can to one row.
- For each cell you just changed, look at its column. That column now has new information.
- Repeat until nothing changes, then sweep the whole grid again.
A disciplined sweep (every row top to bottom, then every column left to right) catches deductions you'd otherwise miss. Most “I'm stuck” moments are a line that changed and never got re-checked.
8. Contradiction (the last resort)
Very hard puzzles occasionally need a step the line techniques can't find. The disciplined approach is to pick one undecided cell and ask, “If this were filled, what follows?” Follow the forced deductions only. If you reach an impossible line, the cell must be empty. It's still pure logic, but it's hard to do in your head, so most puzzle designers avoid requiring it.
Every puzzle in Carve is built so that a line-by-line deduction is always available. You should never need this technique there.
Strategy for 3D nonograms
In a 3D nonogram like Picross 3D or Carve, each clue is a single number on a cube face: the count of cubes in that line that belong to the shape, plus a marker for how many separate groups they form. In Carve a bar means one group, two dots mean exactly two groups, and three dots mean three or more. The techniques adapt like this:
- Zeros first. A 0 means every cube in that line goes, so chip it all. Clearing zeros also exposes new faces and new clues.
- Full lines second. A number equal to the line length means every cube stays. Mark them so you can't chip them by mistake.
- Overlap still works for one-group clues. A bar-marked 4 in a line of 5 always covers the middle three cubes, exactly like the 2D formula.
- Two groups, one short. A two-dot clue that is one less than the line length means there is exactly one gap, and it can't be at either end (or there'd be only one group). Both end cubes stay.
- Spin before you think. The most useful clue is often on a face you're not looking at. Rotate the block and scan every side for zeros and full lines.
- Slice to reach the middle. On bigger blocks the interior cubes are hidden. Carve's slice sliders pull layers aside so you can read and work the inner rows.
The extra dimension means every cube is crossed by three lines instead of two, so cross-checking (technique 7) is even more powerful. A cube proved in one direction instantly updates two other lines.
Frequently asked questions
What is the overlap method in nonograms?
Push a run as far left as it can go, then as far right as it can go. Any cells covered by the run in both positions must be filled. For a single clue k in a line of n cells, that's 2k − n cells in the middle, whenever k is more than half of n.
How do I calculate slack in a nonogram row?
Add up all the clue numbers, add one for each gap between runs, and subtract the total from the line length. The result is the slack. Any run longer than the slack gives you (run length − slack) guaranteed filled cells.
Should I ever guess in a nonogram?
Not in a well-designed puzzle. If you truly run out of line-by-line deductions, test a single cell by asking what would happen if it were filled, and follow the consequences only far enough to hit a contradiction. That's still logic, not guessing.
What's the hardest nonogram technique?
Contradiction (sometimes called probing or trial) is the hardest, because you have to hold a hypothetical chain of deductions in your head. Most published puzzles, including every puzzle in Carve, are designed so you can avoid it.
How do I get faster at nonograms?
Learn to recognise the overlap and slack patterns at a glance, always cross out finished lines immediately, and sweep rows and columns in a fixed order so no line goes unchecked.