TECHNIQUE 3 OF 4
Completed runs: prove the boundaries
Empty marks do real work. Once a run is known to be complete, its boundaries prevent it from growing and may constrain the next run.
The rule
If a filled block is established as one whole clue run, any existing square immediately before or after it must be empty. If all required runs have been placed, every other square in the line is empty. Establish which clue the block belongs to before closing it; a small filled block may still be part of a longer run.
A common mistake
A block of two filled squares is not automatically a completed run of two when the clues include both 2 and 4. Closing it without identifying the run can cut off a valid extension.
TRY IT 1 · nonograms-completed-run-1
| C11 2 | C21 2 | C35 | C44 | C51 2 | |
|---|---|---|---|---|---|
| R12 | |||||
| R21 · 3 | |||||
| R33 | |||||
| R45 | |||||
| R51 · 3 |
What can you justify?
Use the completed runs 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. Other undecided cells in this line are left open.
TRY IT 2 · nonograms-completed-run-2
| C14 | C21 3 | C31 3 | C41 | C51 1 1 | |
|---|---|---|---|---|---|
| R12 · 1 | |||||
| R21 | |||||
| R33 · 1 | |||||
| R43 | |||||
| R55 |
What can you justify?
Use the completed runs technique. What does it prove about R1C4? Treat the displayed starting marks as established facts.
Written explanation
Column 4, clue 1: 1 legal arrangement remains after respecting the current marks. Every arrangement agrees on R1C4 empty, R2C4 empty, R4C4 empty. Other undecided cells in this line are left open.
A two-square block may belong to the longer clue
A twelve-square line has clues 2, 4. Positions 5 and 6 are proven filled; all other positions are unknown.
The tempting move
Treat positions 5–6 as the completed run of 2 and mark positions 4 and 7 empty.
Why it fails
The first run can occupy 1–2 and the four-square run 4–7. That valid arrangement fills both positions you would mark empty. Another valid arrangement uses 5–6 for the short run and 8–11 for the long run, so the supplied block has not been assigned to a clue.
What is actually justified
Do not close the block merely because its current length matches the smaller clue. Establish which run it belongs to, or compare all legal arrangements for forced marks.
Transfer the idea
A seven-square line has the single clue 3. Positions 3, 4 and 5 are already proven filled. Which of the other positions can be marked empty?
Check your reasoning
All of them: 1, 2, 6 and 7. The filled block already supplies the line's only run. Positions 2 and 6 cannot extend it, and positions 1 and 7 cannot form additional runs.