Using Crossed-Out Cells
Empty cells aren't just gaps—they're powerful constraints. Learn how marking crosses unlocks new logic and prevents mistakes.
Why Marking Empty Cells Matters
Beginners often focus only on finding filled cells, treating empty cells as unimportant. This is a mistake. In nonograms, knowing what's empty is just as valuable as knowing what's filled.
When you mark a cell as empty (with an X or dot), you create a boundary. Groups of filled cells cannot cross these boundaries, which limits where they can fit. This constraint frequently reveals new forced cells that weren't visible before.
Golden Rule: Always mark cells you know are empty. Never leave them blank just because they seem "uninteresting." The structure they reveal is critical.
How Crosses Constrain Groups
A crossed cell acts as an impassable barrier for groups. If a group must fit somewhere in a line, and there's a cross in the way, the group must fit entirely on one side of that cross or the other—it cannot span across.
This creates segments: continuous stretches of cells between crosses (or between a cross and the edge). Each segment has a maximum size, and if a group is too large for a segment, it cannot fit there.
Example: Line with crosses
This line has 3 segments: [cells 1-2], [cells 4-6], [cells 8-9]
If this line has a clue of "4", the group must fit in segment 2 (cells 4-6) because that's the only segment large enough. This immediately lets us apply the overlap technique within that segment.
Example 1: Finding New Forced Cells
Consider a row of 10 cells with a clue of "5". Without any crosses, the overlap technique gives us 1 forced cell (the middle one). But what if we mark some cells as empty?
Before marking empties:
Only 1 forced cell from overlap
After marking empties (from column analysis):
Crosses at positions 1, 2, 8, 9, 10
Result:
Now the group of 5 must fit entirely in cells 3-7 (the only segment large enough). Since 5 cells must fit in a space of 5, all 5 are forced!
Without the crosses, we had 1 forced cell. With the crosses, we have 5. This is the power of marking empties—it transforms ambiguous lines into fully determined ones.
Example 2: Separating Multiple Groups
Crosses are especially powerful when dealing with multiple groups. They can tell you which segment each group belongs to, even when overlap alone can't.
Consider a row of 11 cells with clues "3 2".
Some cells already filled:
The cross at position 6 divides the line into two segments:
Left segment [1-5]: has 2 filled cells, needs the "3" group
Right segment [7-11]: has 1 filled cell, needs the "2" group
Completing the groups:
Left segment: fill cell 3 to complete "3", mark 4-5 empty. Right segment: fill cell 7 to complete "2", mark 10-11 empty.
Example 3: Edge Extensions
When a filled cell is adjacent to a cross, you can sometimes extend it to complete a group.
Starting state (clue: "4"):
2 filled cells adjacent to a cross
The group needs 4 cells total. The filled cells at positions 3-4 cannot extend right (blocked by the cross at 5). They must extend left.
Result:
Fill positions 1-2 to complete the group of 4.
This pattern—filled cells next to a cross—is one of the most common triggers for marking new cells. Always check for it.
When to Mark Cells as Empty
Knowing when to add crosses is key to effective solving. Here are the most common scenarios:
1. Completing a Group
When you finish filling a group, mark the cells immediately before and after as empty. This prevents accidentally extending the group and provides boundaries for other groups.
2. Cells Too Far from Any Group
If a cell cannot possibly be part of any group based on position and clues, mark it empty.
3. Information from Intersecting Lines
If a column analysis tells you a cell must be empty, mark it in the row too. This cross-propagation is crucial.
4. All Groups Accounted For
If you've placed all groups in a line, mark all remaining cells as empty.
Common Mistakes to Avoid
❌ Not marking empties at all
Some solvers only mark filled cells. This makes puzzles significantly harder and increases error rates. Always mark empties.
❌ Forgetting to mark the boundaries of completed groups
When you complete a group, immediately mark the adjacent cells as empty. Forgetting this step loses valuable information.
❌ Marking empties based on guesses
Only mark a cell empty if you can prove it through logic. Guessing leads to contradictions later.
The Workflow: Overlap + Crosses
In practice, you'll use overlap and crosses together in a cycle:
Apply overlap to find initial forced cells
Mark boundaries around completed groups with crosses
Re-apply overlap within the new segments created by crosses
Mark new empties based on the updated layout
Repeat until the line is complete
This cycle is the heartbeat of nonogram solving. Overlap finds filled cells, crosses create structure, and structure enables more overlap.