TECHNIQUE 4 OF 4
Crossing information: let rows and columns help each other
A difficult-looking row may become simple after one column deduction. Revisit the lines touched by a new mark instead of repeatedly scanning the whole puzzle unchanged.
The rule
Every cell belongs to both a row and a column. Transfer a proven filled or empty mark to the other line, then discard any arrangements that disagree with it. Continue only when all remaining arrangements agree on a new square. A transferred mark must itself be justified; repeating a guess in another line does not turn it into evidence.
A common mistake
Using only filled squares misses deductions created by empties. Also, a line with no legal arrangements is a contradiction to investigate, not permission to ignore whichever clue is inconvenient.
TRY IT 1 · nonograms-crossing-1
| C11 2 | C21 2 | C35 | C44 | C51 2 | |
|---|---|---|---|---|---|
| R12 | |||||
| R21 · 3 | |||||
| R33 | |||||
| R45 | |||||
| R51 · 3 |
What can you justify?
Use the crossing information technique. What does it prove about R1C1? Treat the displayed starting marks as established facts.
Written explanation
Column 1, clue 1, 2: 1 legal arrangement remains after respecting the current marks. Every arrangement agrees on R1C1 empty, R3C1 empty, R4C1 filled, R5C1 filled. Other undecided cells in this line are left open.
TRY IT 2 · nonograms-crossing-2
| C14 | C21 3 | C31 3 | C41 | C51 1 1 | |
|---|---|---|---|---|---|
| R12 · 1 | |||||
| R21 | |||||
| R33 · 1 | |||||
| R43 | |||||
| R55 |
What can you justify?
Use the crossing information technique. What does it prove about R2C2? Treat the displayed starting marks as established facts.
Written explanation
Column 2, clue 1, 3: 1 legal arrangement remains after respecting the current marks. Every arrangement agrees on R2C2 empty, R3C2 filled, R4C2 filled, R5C2 filled. Other undecided cells in this line are left open.
Conflicting deductions do not strengthen each other
One five-square row has clues 1, 1. Two earlier column decisions have left its positions 2 and 3 marked filled; the other positions are unknown.
The tempting move
Accept both transferred marks and treat them as the row's two required filled squares.
Why it fails
Clues 1, 1 require two separate single-square runs with at least one empty square between them. Adjacent filled positions 2 and 3 form a run of 2, so no arrangement matches the row's clue and both marks.
What is actually justified
The current constraints conflict. Revisit the earlier decisions or copied clues. This row proves that both filled marks cannot be correct together; it does not by itself identify which one should be removed.
Transfer the idea
A five-square column has clue 3. A row deduction proves that its first square is empty. Which squares are now forced filled in that column?
Check your reasoning
The run can now occupy only positions 2–4 or 3–5, so positions 3 and 4 are forced filled. Position 1 stays empty, while positions 2 and 5 remain undecided until more information arrives.