Advanced solving techniques

Once you know the rules, the difference between a solver who stalls on a hard puzzle and one who finishes it is a small set of repeatable deductions. None of them require luck. A well-constructed nonogram — including every puzzle on this site — has exactly one solution that is reachable by pure logic, so guessing is never required and never helps.

If you are new to the puzzle, read the beginner's guide first, and keep the glossary handy for any unfamiliar terms. In the illustrations below, # is a known-filled cell, . is an unknown cell, and X is a cell you have proven empty.

1. The overlap method

This is the single most productive technique on large grids. When a run is long relative to its line, the run cannot avoid covering the middle cells no matter where it sits. Slide the run as far left as it will go, then as far right; any cell filled in both extremes is filled in the answer.

There is an exact formula. For a single clue of length k in a line of length n, the run can start anywhere from the left edge to position n − k. The cells that are covered in every one of those positions are the overlap, and there are exactly 2k − n of them, centered in the line. Whenever 2k − n is positive, you can fill that many cells immediately.

Line of 10, clue "8"  (k = 8, n = 10)

leftmost:   # # # # # # # # . .
rightmost:  . . # # # # # # # #
overlap:    . . # # # # # # . .   <- 2k - n = 6 cells filled

The method extends to multi-clue lines: pack all runs to the left, pack them all to the right, and overlap each run against its own leftmost and rightmost copy. A clue of 4 4 in a width of 10 leaves almost no slack, so most of both runs is forced.

2. Edge logic

Walls and known-empty cells behave the same way: they are hard boundaries a run cannot cross. The first clue in a line is anchored to the left wall, the last clue to the right wall, and this often lets you place cells even when the overlap formula gives nothing on its own.

The classic case is a filled cell touching the edge. If the line starts with clue 3and the very first cell is already filled, that run must begin at the wall — so the next two cells are filled and the cell after the run is empty.

Clue "3", first cell known filled:

before:  # . . . . . . .
after:   # # # X . . . .   <- run anchored to the wall

A proven-empty cell acts as a movable wall: treat the segment between two X marks as a tiny independent line and apply edge logic and overlap inside it.

3. Gap splitting and forcing

A single known-empty cell can split a line into two segments. When it does, ask which clues can still fit in each segment. Often a run is simply too big for one side, which forces it entirely onto the other — and that placement frequently fills several cells at once.

Clue "4" in width 7, cell 4 known empty:

. . . X . . .
^^^^^       ^^^   left segment = 3 wide, right = 3 wide

A run of 4 fits in neither segment alone...
...so the puzzle is only consistent if you re-examine:
the "4" cannot exist here -- the X disproves it,
forcing the run into whichever segment can hold it.

The same idea handles ordering. If the left segment is only wide enough for the first clue and the right segment is the only place the second clue can sit, the assignment is forced and you can solve both segments independently. Splitting turns one hard line into two easy ones.

4. The contradiction test

When direct deduction stalls, prove a cell by elimination. Tentatively assume the cell is filled, propagate the consequences through its row and column as far as they go, and watch for an impossible state — a clue that can no longer fit, a run with no room, or a count that overshoots the line. If the assumption forces a contradiction, the cell must be empty. The reverse works too: assume empty, reach a contradiction, conclude filled.

This is still pure logic, not guessing. You are not committing to a guess and hoping; you are performing a short proof and keeping only the result. Crucially, propagate only forced moves while testing, so the chain stays valid. On the puzzles here you will rarely need more than a one- or two-step look-ahead, because the construction guarantees a simpler path exists.

Suppose this cell is filled:

. . [#] . .   -> its column clue "1" is already satisfied
              -> but that forces the crossing row's "2 2"
                 to overflow the remaining cells -> impossible
Therefore the cell is empty:  . . X . .

5. Working the clue numbers from both ends inward

A line's clue list is read left-to-right, but you do not have to solve it in that direction. The first number is pinned to the left wall and the last number is pinned to the right wall, so you can make progress from both ends and meet in the middle. Anything you fix on the right tightens the room available on the left, and vice versa.

A useful habit is to compute the “minimum span” of a line: the sum of all clue lengths plus one gap between each pair. Subtract that from the line length to get the total slack. The smaller the slack, the less each run can move — and when the slack is zero, the entire line is determined. Knowing the slack tells you, before you fill anything, how much the overlap method will hand you from each end.

6. Using completed lines to constrain crossing lines

Every cell lives in both a row and a column, and that shared membership is the engine of the whole puzzle. The moment you fill or empty a cell, switch your attention to the crossing line and ask what just changed there. A single new filled cell can anchor a run; a single new X can split a line and trigger the forcing logic above.

When a line is fully solved, it stops being a puzzle and becomes a source of fixed clues for every line it crosses. Sweep its filled and empty cells into the perpendicular lines and you will usually unlock a fresh wave of deductions. Cycling between rows and columns this way — row pass, column pass, repeat — resolves nearly every grid without a single guess. If you ever feel stuck, the next deduction is almost always hiding in the line that crosses the one you are staring at.