How to solve nonograms, step by step · blobsmith.lbwma.com/how-to-solve-nonograms · ten free printable nonogram sheets with the answer key: blobsmith.lbwma.com/free-nonograms
A nonogram — also sold as picross, griddler, hanjie or paint by numbers — looks like guesswork the first time and isn't. There are four deductions, and they are enough. Each one below has a drawn example, and at the end a real 15×15 puzzle opens pass by pass until it turns into a picture.
The numbers on a row (or column) give the lengths of the runs of filled cells in that line, in the order they appear. Between two neighbouring runs there is at least one empty cell.
That's all of it. A row of 10 with the clue 3 4 holds a run of 3, then a gap, then a run of 4 — and working out where they sit is the whole game. When the grid closes, the filled cells make a picture: that's how you know you were right before you look at the key.
One convention saves an enormous number of mistakes: put an × in every cell you have proved is empty. In the diagrams on this page, a black cell is filled, an × is confirmed empty, and a blank cell is still unknown.
Before you think about anything, sweep the grid for a row or column whose numbers leave no choice. The extreme case is a clue equal to the length of the line:
The same holds when the clues add up. In a row of 10, the clue 4 5 needs 4 + 1 + 5 = 10 cells — it fits exactly one way. Every line like this is free fuel for the columns crossing it, and it is where every puzzle should start.
This is the most important technique and the least obvious. Push the run as far left as it will go, then as far right: any cell filled in both positions is filled for certain — and you still don't know where the run really starts.
The arithmetic is 2k − n: a clue of k in n free cells. In a row of 10:
| clue | cells guaranteed | arithmetic |
|---|---|---|
| 5 | — | none: the block still slides too far |
| 6 | 2 | 2×6 − 10 = 2 |
| 7 | 4 | 2×7 − 10 = 4 |
| 8 | 6 | 2×8 − 10 = 6 |
| 9 | 8 | 2×9 − 10 = 8 |
| 10 | 10 | 2×10 − 10 = 10 |
Which is why you work the biggest numbers first. Below half the length of the line, overlap gives you nothing at all, and you are spending attention where there is no information to find.
One filled cell near an edge is often worth the entire line. If the first cell is filled and the clue is 4, the run can only start there:
Keep the principle rather than the case: a run that touches a wall or an × loses all its freedom. That is why hunting along the edges pays better than attacking the middle of the grid.
Knowing where a run cannot go is half the game, and it is the step nearly every beginner skips. In the same row of 10 with the clue 3 4, an empty row decides 3 cells. Mark one cell as empty and it goes to 6:
Every × you write into a row arrives as new information in the column that runs through it. That exchange is what makes a grid collapse.
A real 15×15 nonogram — and it is one of the ten in our free pack, the alien, so you can print the sheet and follow along with a pencil. A pass here means sweeping every row or every column once, applying the four deductions above and nothing else, alternating — which is exactly what you do at the table. Cells the pass has just opened are drawn in blue.
Look at the shape of the work. The first pass over the rows alone decides 120 of 225 cells — nearly all of that is plain overlap, deduction 2, without looking at a single column. The next pass, over the columns, feeds on what the rows just wrote and opens 65 more. After these 4 the rest falls out on its own; the grid closes in 5 passes, and the picture appears.
So when you get stuck, it is almost always because you stopped alternating: you solved everything the rows had to give and kept staring at the rows. The new information is in the column crossing the last cell you filled.
Some nonograms do not close on logic: at some point the puzzle demands that you pick one of two possibilities and backtrack if it goes wrong. Nothing on the sheet warns you. You find out at your kitchen table half an hour in, and it is the standard defect of puzzles generated in bulk and dumped into a PDF.
Worth knowing before you start doubting your own reasoning: in a sound puzzle there is always a deducible cell somewhere on the grid. If you have checked everything and there isn't one, the puzzle is broken, not you.
Ours go through a solver before they become a page — the same one that drew the steps above. It rejects a puzzle if it does not close on row-and-column logic alone, if a second grid satisfies the same numbers, or if the solution isn't exactly the picture that was drawn. Puzzles that fail are thrown away, not patched. All 10 of the 10 in the free pack below pass it.
The same ten sheets ranked by measured difficulty — passes to close, cells opened by the first sweep — with a 45-minute plan built on those numbers: nonograms in the classroom
Type its clues into the nonogram solver and it works the grid the same way this page teaches — and if the clues are impossible it names the row or column that proves it, which is usually a typo rather than a hard puzzle.
Drawing your own puzzle instead of solving one? The nonogram checker reads the clues off your picture and says whether the four rules above are enough to finish it, marking the squares that would force a guess.
The same guide, written for Brazil: como resolver nonograma · 10 nonogramas grátis