Nonogram clues do more than specify block lengths. They also fix the order of those blocks. That order creates an earliest and latest legal position for every clue block, even before you know its exact placement.
Example: clues 2 3 on a ten-cell line
Take an empty 10-cell line with clues 2 3.
The first block can start as early as cell 1. But it cannot start arbitrarily late: it must leave at least one separator and enough room for the later block of three.
Its latest start is cell 5, so its possible reach is cells 1–6.
The second block cannot start before cell 4 because the first block of two plus a mandatory separator must come first. Its latest start is cell 8, giving it possible reach through cell 10.
Nothing is necessarily forced filled yet, but the two blocks already have different legal territories because clue order prevents them from swapping.
Earliest placement
A block's earliest placement is the farthest left or highest it can appear while all earlier clues, separators, known filled cells, and X marks remain satisfiable.
For later blocks, this bound depends on everything that must precede them.
Latest placement
A block's latest placement is the farthest right or lowest it can appear while still leaving enough legal space for every clue that follows.
For earlier blocks, this bound depends heavily on the blocks after them.
Why order matters on multi-clue lines
If a line contains clues 2 5 3, those are not three interchangeable blocks. The 2 must come first, the 5 second, and the 3 last.
A filled fragment near the right edge might physically fit inside a block of length 2, but it cannot be clue 2 if doing so would leave no place for clues 5 3 afterward.
This is why block reach must be computed with clue identity and order, not just block length.
Worked example: known cells tighten three ordered block ranges
Take a 12-cell line with clues 2 4 2. Cell 4 is confirmed empty and cell 8 is confirmed filled:
???×???■????
Exactly six complete line patterns remain compatible with those states.
Across those six patterns, the legal start positions are:
| Clue block | Earliest start | Latest start |
|---|---|---|
first 2 | 1 | 2 |
middle 4 | 5 | 6 |
final 2 | 10 | 11 |
The block identities matter. We are not asking where “a 2-block” can go; we are asking where the first and last 2-blocks can go while the middle 4 remains between them.
Those bounds immediately imply several consensus cells:
- cell 2 is covered by the first block in every valid pattern;
- cells 6–8 are covered by the middle block in every valid pattern;
- cell 11 is covered by the final block in every valid pattern;
- cell 4 remains empty by the given constraint.
So bounds are not merely descriptive ranges. Once they become narrow enough, they expose overlap for each ordered block.
Bounds are coupled, not independent intervals
It is tempting to compute a range for each clue and then treat those ranges separately. On multi-clue lines, that can be wrong.
Moving one block later may force the next block later because a mandatory separator must remain between them. Moving the final block earlier may pull the previous block earlier for the same reason.
Think of the ordered clue blocks as a chain of coupled ranges:
- earlier blocks push later blocks to the right/down;
- later blocks push earlier blocks to the left/up;
- X marks and segment boundaries can cut the chain into smaller subproblems;
- known filled fragments can pin one block and tighten both neighbors.
This is why recalculating bounds from both ends is more reliable than estimating each clue's reach independently.
Bounds versus full valid-pattern analysis
Two lines can have identical earliest/latest bounds yet differ in which intermediate combinations are legal. Extreme placements compress information, so they can miss deductions that only appear when every full pattern is considered.
Use bounds when they give a clean proof. Move to valid line patterns when the extremes leave too much ambiguity or when known cells create non-convex placement possibilities.
Bounds tighten as the line is solved
Every confirmed state can change the ranges:
- an X may push a block inward;
- a filled fragment may force a block to cover a particular area;
- a completed block may fix one clue's identity and split the remaining sequence;
- segment assignment may restrict a clue to a smaller region;
- crossing-line deductions may remove placements that were previously possible.
Recalculate bounds whenever a line gains important information.
Bounds create overlap
Once you know the earliest and latest legal placement for the same clue block, compare them.
Cells occupied by that block in every legal placement are forced filled. This is the deeper version of the overlap method introduced earlier.
Bounds can also create forced empties
If a cell lies outside the legal reach of every clue block, it is forced empty.
So placement bounds support both sides of line solving:
- intersection of a block's legal placements → forced fill;
- outside the union of all block reaches → forced empty.
Step-by-step bounds analysis
- keep clue identities in their fixed order;
- place all earlier clues as early as legally possible to find a block's earliest bound;
- place all later clues as late as legally possible to find its latest bound;
- respect existing X marks and filled fragments in both calculations;
- compare the legal range of each block;
- mark only states shared by every valid placement;
- repeat after new deductions.
This is closely related to the left-most/right-most line-solving approach used by computer Nonogram solvers.
Common mistakes
Treating same-sized clues as interchangeable
Even two clues with the same number have different identities because their order is fixed.
Computing a bound without reserving space for later clues
A block cannot use cells that make the rest of the clue sequence impossible.
Using only the extreme start cells
The full block occupies several cells. Bounds describe ranges for the entire block, not just its first cell.
Filling every cell between earliest and latest reach
That interval represents possible occupancy, not guaranteed occupancy. Use intersection for forced fills.
What to learn next
Placement bounds are a compact way to represent many possibilities. Valid line patterns goes one step further by listing or reasoning over every complete arrangement that satisfies the line's clues and known states.
FAQ
Are earliest and latest placements enough to solve every line?
No. Extreme-placement methods are powerful but not complete in every constrained line. Some deductions require considering the full set of valid patterns.
Do bounds apply to rows and columns equally?
Yes. “Earliest” means leftmost for rows and uppermost for columns; “latest” means rightmost or lowermost.
Can a block's bounds become a single placement?
Yes. If earliest and latest placement coincide, the block's exact position is fixed.