Most players learn one trick for starting a nonogram row: push the blocks as far left as they go, push them as far right as they go, and paint whatever overlaps. It is a real rule and it works. It is also a small special case of something stronger, and on a 20 by 20 board the gap between the two is enormous. This page sets out the general method, then shows what each version is worth measured against the trophy grid in Nonogram.
The method in one sentence
For a single line, list every arrangement of its blocks that fits the line's length and agrees with the cells you already know, then fix every cell that comes out the same in all of them. If a cell is painted in every surviving arrangement, paint it. If it is empty in every surviving arrangement, it is empty. If the arrangements disagree about it, leave it alone and move on. Nothing else is ever needed, and nothing weaker is ever safe.
The two halves of that sentence carry all the weight. Every arrangement means the deduction is complete: you cannot squeeze more out of that line right now, so there is no reason to keep staring at it. Agrees with the cells you already know means the deduction gets stronger every time a crossing line reports back. That second half is the part the memorised overlap rule throws away.
What the memorised rule leaves on the table
Run the left-push, right-push overlap rule across all twenty rows and all twenty columns of the trophy grid, on a blank board, and it fixes 148 cells out of 400, or 37.00 percent. Then it stops. It has to stop, because it never reads the board: feed it the same clues a second time and it produces the same 148 cells. It also produces nothing but paint. Of the 190 empty cells in the finished picture it identifies zero, because the rule has no way to express "no block can reach here".
Full intersection, over exactly the same forty lines and the same blank board, fixes 296 cells out of 400 on the very first sweep. That is 74.00 percent, and 148 more cells than the shortcut, from the same clues and the same amount of looking. Two more sweeps close the remaining 104 cells and the grid is finished.
| Approach | First sweep | Empty cells found | Where it ends up |
| Memorised overlap | 148 of 400 (37.00%) | 0 of 190 | Stalls at 148 |
| Full intersection | 296 of 400 (74.00%) | Included in the 296 | 400 of 400 in three sweeps |
Worked Example: three lines, three different lessons
Row 5, clue 3 12 3. The blocks need 18 cells and the two mandatory gaps between them need 2 more, which is exactly the width of the row. Only one arrangement fits. The overlap rule gets 18 painted cells here and does well, but it stops there. Intersection returns all 20 cells: the same 18 painted, plus the two gaps at columns 4 and 17 proved empty. Those two empty cells are not decoration. They are what later tells column 4 that its top block ends where it ends.
Column 1, clue 2. A single block of two in a column of twenty allows 19 positions, and no cell is painted in all 19, so both the overlap rule and intersection return nothing on a blank board. The overlap rule never returns to it. Intersection does, because the moment rows 1, 5, 6 and 20 are solved, column 1 knows four of its cells: two empty at the ends, two painted at rows 5 and 6. Exactly one of the 19 arrangements survives that, and the column settles 16 new cells at once, every one of them empty. A clue worth nothing became worth 16 cells with no new clue information at all, only new board information.
Column 9, clue 18. A block of 18 in a column of 20 overlaps into the middle 16 cells, which the shortcut finds. But rows 1 and 20 are already known to be empty from their clue of 0, and a block of 18 cannot start at row 1 or end at row 20 in that case. One arrangement survives: rows 2 through 19. Intersection returns 18 painted cells, two more than the shortcut, and the difference is exactly the two cells the shortcut could never see because it refuses to look at the board.
Why "all arrangements" is cheaper than it sounds
The obvious objection is workload: listing every arrangement sounds like something only a computer would do. In practice the count collapses fast, because the board keeps tightening it.
On the blank trophy grid the forty lines together admit 2,640 arrangements. The two worst are columns 4 and 17, whose clue of 2 2 2 allows 455 each, more than a third of the total between them. Rows 5 and 6, meanwhile, allow exactly one apiece and columns 9 to 12 allow three each. That distribution is what makes the method practical: the lines worth starting with are the ones with almost nothing to enumerate.
You never have to enumerate the hard lines cold. By the time the grid reaches columns 4 and 17, most of their cells are already known, and the number of arrangements that agree with what is on the board has collapsed from 455 to a handful. That is the practical rule the method gives you for free: work the lines with the fewest surviving arrangements, always. Not the ones that look interesting, and not the ones you happen to be next to.
How the yield moves as the grid fills
Sweep the rows, then the columns, then repeat, and the returns fall away in a very steep curve:
| Sweep | Cells from rows | Cells from columns | Total | Board settled |
| 1 | 156 | 140 | 296 | 74.00% |
| 2 | 70 | 22 | 92 | 97.00% |
| 3 | 12 | 0 | 12 | 100.00% |
| 4 | 0 | 0 | 0 | 100.00% |
Notice what the columns do. On the first sweep they supply 140 cells, nearly as many as the rows. On the second they supply 22, and on the third none at all: the rows finish the job because the columns have given up everything they had. The fourth sweep only confirms that both directions are exhausted.
Doing it by hand without losing your place
The method translates into a short routine that needs no arithmetic beyond addition:
- Add up the clue numbers plus one gap between each pair of blocks. If the total is 20, the line has one arrangement and is finished immediately. Rows 5 and 6 qualify, at 20 cells each.
- If the line has a clue of 0, the whole line is empty. That is 20 cells for no work, and on this board rows 1 and 20 both qualify.
- Otherwise, look at what the line already knows. Every painted cell rules out arrangements that leave it empty; every empty cell rules out arrangements that cover it. Keep only what survives.
- Paint what all survivors agree is painted, mark what they all agree is empty, and leave the rest.
- Move to a crossing line that just gained a cell. Never re-open a line that gained nothing since you last read it.
Step 5 saves the most time. Seventeen of the forty lines return nothing on a cold read, so sweeping in fixed order spends much of the first pass on lines that cannot pay yet. Following the freshly-updated lines instead closes the grid in 41 line reads, 21 rows and 20 columns.
Frequently asked questions
Is intersection different from the overlap rule, or just a longer version of it?
It contains the overlap rule as its simplest case and goes past it in two ways: it uses the cells already on the board, and it identifies empty cells as well as painted ones. On this grid those two additions are the difference between 148 cells and the whole 400.
Do I need to write down every arrangement?
No. Check whether the line has one surviving arrangement, which is quick, and otherwise slide each block to its extreme positions given the cells already known. A line with hundreds of possibilities is the wrong line to work on.
Which lines should I read first on this board?
The ones whose clue barely fits. Rows 5 and 6 have exactly one arrangement, rows 1 and 20 have one by definition, and columns 9 to 12 have three each. Those eight lines settle a large share of the opening with no board information at all.
Why do empty cells matter so much if the game never checks them?
Because they are what makes the next line's intersection collapse. Column 1 gains 16 cells only because rows 1 and 20 were proved empty and rows 5 and 6 were proved painted. Whether you record the empty cells with an X or hold them in your head, the deduction depends on them.
Can this method ever fix a cell wrongly?
Not if you apply it as stated. A cell is fixed only when every arrangement consistent with the clue and with the known cells agrees on it, so any solution of the puzzle must agree too. That is why nothing you deduce this way ever has to be undone.
What if two lines seem to contradict each other?
On this grid they cannot, and the reason is arithmetic you can check yourself: the row clues sum to 210 and the column clues sum to 210, the same 210 cells counted along both axes. If you ever reach a contradiction here, an arrangement was discarded that should have been kept, so re-read the line rather than the clue.
Related guides
- How to Play Nonogram for the rules and the standard techniques stated in general form.
- The trophy walkthrough for this method applied line by line from an empty board.
- Nonogram no-guess strategy for the argument that intersection alone is always enough here.
- How to Solve Sokoban Puzzles and How to Play Minesweeper for the same discipline on other boards.
- Best puzzle games for the rest of the deduction shelf.