Overlap & Forced Cells
The foundational technique for starting any nonogram. Learn how to identify cells that must be filled, regardless of where groups are positioned.
What is the Overlap Technique?
The overlap technique answers a simple question: if a group of filled cells can fit in multiple positions along a line, are there any cells that would be filled no matter where we place the group?
When a group is large enough relative to the line length, the answer is yes. Some cells in the middle will be filled in all valid positions, making them forced cells—cells you can mark with 100% certainty.
This is usually your first move in any nonogram puzzle, and it's the technique you'll use most frequently throughout solving.
The Basic Formula
Forced cells appear when:
Group length > (Line length ÷ 2)
More precisely, the number of forced cells is:
Forced cells = (Group length × 2) − Line length
But you don't need to memorize formulas. The visual logic is intuitive: place the group at the leftmost position, then at the rightmost position. Any cell filled in both cases is forced.
Example 1: Single Group in a Row
Consider a row of 10 cells with a clue of "7". Where can the group of 7 fit?
Leftmost position:
Rightmost position:
Overlap (forced cells):
Cells 4, 5, 6, and 7 are filled in both cases, so we can mark them immediately. That's 4 forced cells from a group of 7 in a line of 10.
Quick check: (7 × 2) − 10 = 14 − 10 = 4 forced cells ✓
Example 2: Multiple Groups
The technique works with multiple groups too, but you must account for the required gaps between them.
Consider a row of 9 cells with clues "3 2". This means:
A group of 3 filled cells
At least 1 empty cell
A group of 2 filled cells
Total minimum space needed: 3 + 1 + 2 = 6 cells
Leftmost arrangement:
Rightmost arrangement:
Overlap:
Cell 3 is forced from the first group (3 cells). Cells 5, 6, and 8 are forced from the overlap regions. But note: we don't yet know which group cells 5-6 belong to! That requires additional logic.
Example 3: When There's No Overlap
Not every line produces forced cells. If the groups are small relative to the line length, there may be no overlap.
Consider a row of 10 cells with clues "2 2". Minimum space: 2 + 1 + 2 = 5 cells. Since 5 ≤ 10 ÷ 2, there's plenty of room to move the groups around.
One possible arrangement
Another possible arrangement
These two arrangements don't share any filled cells, so we can't mark anything yet. This line requires other techniques or information from intersecting lines before we can make progress.
When to Use This Technique
1.
At the start of every puzzle
Scan all rows and columns for large groups. Mark any forced cells you find. This gives you a foothold to begin solving.
2.
After marking any new cells
When you fill or cross out cells, revisit the affected lines. The newly marked cells may constrain group placement enough to create new forced cells.
3.
When you're stuck
If other techniques aren't yielding progress, do another full pass looking for overlap. Sometimes a small change elsewhere unlocks new forced cells.
Common Mistakes to Avoid
❌ Mistake: Marking cells outside the overlap region
Only mark cells that are filled in all possible positions. If a cell is filled in some positions but not others, it's not forced yet.
❌ Mistake: Forgetting the required gaps between groups
When calculating positions for multiple groups, always leave at least one empty cell between them. Failing to account for gaps leads to invalid overlap calculations.
❌ Mistake: Assuming forced cells belong to a specific group
With multiple groups, forced cells might not tell you which group they're part of. You've confirmed they're filled, but group assignment may require additional logic.
Practice Exercise
Try applying the overlap technique to these lines. Which cells are forced?
Line length: 8, Clue: "6"
Answer: Cells 3, 4, 5, 6 are forced (4 cells)
Line length: 12, Clue: "4 3"
Answer: No forced cells (groups are too small)
Line length: 7, Clue: "7"
Answer: All 7 cells are forced (perfect fit)