What Is Line Solving?

Line solving is the foundational technique in nonogram solving: analyzing each row and column in isolation, using only its own clue and the cells already filled or marked in that line, to deduce additional cells. It is called "line solving" because each deduction is made within a single line — no information from other lines is used directly.

Despite working line by line, line solving is not isolated. Every deduction in one line becomes a constraint for every intersecting line. The lines are solved in a continuous loop until no more deductions are possible within any single line — at which point the puzzle is either complete or requires cross-line techniques.

The Core Algorithm

The line-solving algorithm for a single line works as follows:

1. List the remaining unsatisfied blocks and the available space (cells not yet marked X).
2. For each block, calculate its leftmost possible start position and its rightmost possible start position within the available space.
3. Any cell covered by the block in both extreme positions must be filled. (This is the overlap method.)
4. Any gap in the available space that is too small to contain any remaining block must be marked empty.
5. If a block's leftmost and rightmost positions are identical, the block is fully determined: fill all its cells and place X marks on each side.

Apply this algorithm to every line in the puzzle. After each pass, some cells change state, which may allow further deductions in other lines. Repeat until no changes occur.

Worked Example: Single Block

Line: 10 cells, clue: 7

Leftmost position: block covers cells 1–7. Rightmost position: block covers cells 4–10. Overlap: cells 4–7 (4 cells). Fill cells 4, 5, 6, 7 immediately.

No cells are eliminated yet, but 4 cells are confirmed. If another line later reveals that cell 1 must be empty, the leftmost position shifts to cell 2, the rightmost stays at 10, and the overlap expands to cells 4–9 — two more cells can now be filled.

Worked Example: Multiple Blocks

Line: 10 cells, clue: 3 2

Minimum space: 3 + 1 + 2 = 6 cells. Slack: 10 − 6 = 4 cells.

Block 1 (size 3): leftmost starts at cell 1, rightmost starts at cell 5. Overlap on cells 5–7? No — leftmost covers 1–3, rightmost covers 5–7. No overlap for block 1 (slack ≥ block size).

Block 2 (size 2): leftmost starts at cell 5, rightmost starts at cell 9. Overlap on cells 9–10? No — leftmost covers 5–6, rightmost covers 9–10. No overlap for block 2 either.

No cells can be filled in this line from line solving alone. But once an intersecting column reveals that, say, cell 2 is filled, you know block 1 must cover cell 2 — constraining it to start at cells 1, 2, or 3 at most. The rightmost position shifts, and overlaps emerge.

When Line Solving Is Sufficient

For most easy and medium nonograms, line solving applied iteratively until convergence is sufficient to solve the entire puzzle. Well-constructed puzzles at these levels are designed so that each cell follows logically from line solving alone.

Hard and expert puzzles may have regions where line solving stalls — no line can make progress in isolation. At this point, cross-line techniques (working with the combined information from multiple lines simultaneously) become necessary.

Accelerating Line Solving

Line solving speed comes from recognizing high-yield lines quickly. Look for:

• Lines where minimum required space is close to or equal to line length.
• Lines with a single large block (high overlap potential).
• Lines with many cells already filled from previous rounds.
• Lines with cells already near the edge (edge anchoring).

Start each solving session with these lines. They provide the most information per unit of time and cascade into further deductions in intersecting lines.

Practice

Line solving is the only technique you need for easy puzzles. Practice it there until it is automatic, then apply it as the base layer of your approach on medium, hard, and expert puzzles, layering additional strategies on top when line solving alone stalls.