A clue block's reach is the set of cells that block can still legally occupy under the current line constraints. If a cell lies outside the reach of every unresolved clue block, that cell cannot be filled and is therefore confirmed empty.
This is the forced-empty counterpart to overlap: overlap asks where a block must be; reach asks where a block may be.
Example: one clue anchored by a filled cell
Take a 10-cell line with a single clue 4 and a confirmed filled cell at position 5.
Because there is only one clue block, the block must include cell 5.
A four-cell block containing cell 5 can start at positions 2, 3, 4, or 5. Across those placements, it can reach cells 2–8.
That means cells 1, 9, and 10 are outside the block's reach. Since there is no other clue block in the line, those cells are forced empty.
Cell 5 is forced filled because it is already known; cells within 2–8 are not all forced filled. Most of them are merely possible block cells.
Reach is a union; overlap is an intersection
This distinction is worth memorizing:
- reach / union = every cell the block occupies in at least one legal placement;
- overlap / intersection = cells the same block occupies in every legal placement.
Cells in overlap are forced filled.
Cells outside the reach of every block are forced empty.
Cells inside reach but outside overlap remain unresolved unless another deduction settles them.
Why filled anchors reduce reach
At the start of an unconstrained one-clue line, a block may be able to reach almost every cell.
Once a filled cell is known to belong to that block, every legal placement must cover the anchor. That can sharply shrink the block's range and make distant cells impossible.
Confirmed X marks can shrink reach further by forbidding placements that cross them.
Multi-clue lines require all blocks to be considered
A cell outside the reach of one clue block is not necessarily empty. Another clue block may still be able to occupy it.
To mark a cell empty safely, you need to know that no valid assignment of any unresolved clue block can cover that cell.
This is where clue order, segment assignment, completed blocks, and placement bounds become important.
Historical “Simple Spaces” terminology
Older Nonogram explanations often call forced-empty reasoning based on block range Simple Spaces.
VeyraPlay uses block reach and unreachable cells because the wording makes the proof explicit:
- determine where blocks can reach;
- eliminate cells that none can reach.
The historical phrase remains useful as an alias for readers who encounter it elsewhere.
Unreachable cells can split a line
Once an unreachable cell is marked X, it may divide a large open region into smaller segments.
That new separator can then:
- make a gap too small for a remaining clue;
- isolate a confirmed fragment;
- force a clue into one segment;
- create an exact fit;
- strengthen overlap.
So a reach deduction often produces a cascade of simpler deductions afterward.
Step-by-step reach analysis
For an identified clue block:
- determine all legal placements that satisfy current filled cells and X marks;
- take the union of the cells occupied across those placements;
- call that union the block's current reach;
- repeat for other unresolved blocks that might occupy the region;
- mark a cell empty only when it lies outside every relevant block's reach;
- re-scan the line after adding the new X marks.
For simple examples you can do this visually. Advanced solvers may represent each block with explicit earliest/latest bounds or enumerate valid line patterns.
Block reach can change after every deduction
Reach is not fixed by the original clue alone.
A new X can remove several placements. A new filled cell can anchor a block. A completed clue can disappear from the unresolved set. A crossing-line deduction can change which segment a block must occupy.
Whenever the line gains meaningful information, reconsider its block ranges.
Common mistakes
Filling every cell inside the reach
Reach means possible, not certain. Only overlap or another proof gives forced fills.
Marking a cell empty because one block cannot reach it
Check whether another unresolved block could legally occupy the cell.
Ignoring clue order
On multi-clue lines, earlier and later blocks cannot swap positions. Their ordered bounds affect reach.
Treating a filled fragment as belonging to a clue without proof
If block identity is ambiguous, the calculated reach may be for the wrong clue.
What to learn next
Block reach creates many of the X marks used by gap elimination. Once the line contains more confirmed empty cells, you can split it into segments and determine which clue blocks can fit in each region.
FAQ
Is every cell outside one block's reach empty?
Only on a one-clue line or when you know no other unresolved block can occupy it. On multi-clue lines, consider all relevant blocks.
Is block reach the same as a block's earliest-to-latest interval?
Often that interval is a useful approximation or representation, but existing separators and complex constraints can make legal placements more structured. The core definition is the union of all legal cells for the block.
What is “Simple Spaces”?
It is a historical/community name for related forced-empty reasoning. VeyraPlay's canonical explanation uses block reach and unreachable cells.