How to Solve Nonograms: A Checked 5×5 Example

How do you solve a nonogram?

Solve a black-and-white nonogram by matching each row and column to its ordered clues. Each number describes a consecutive block of filled cells, with at least one empty cell between blocks. Mark only cells forced by the clues, distinguish confirmed empty cells from unknown ones, and alternate between rows and columns. Check every completed line rather than guessing what the picture should look like.

The original 5×5 puzzle below has one verified solution. It is a practice problem, not an intelligence or cognitive-health assessment. Use notes, enlarged text, a printed grid or any other format that makes the puzzle comfortable to work with. There is no timer to impress.

What do the numbers mean?

In a black-and-white nonogram, a clue of 3 requires one uninterrupted block of three filled cells. A clue of 1, 1 requires two separate single filled cells, in that order, with at least one empty cell between them. It does not mean a joined block of two.

Read row clues from left to right and column clues from top to bottom. The clues describe every filled block in that line, not merely some of them. Cells outside those blocks are empty. These are the black-and-white rules described by puzzle publisher Conceptis.

This lesson does not use colored clues. Color variants can change the separation rule: Conceptis allows differently colored neighboring blocks to touch. Do not import that exception into the black-and-white exercise.

We will use three written marks:

• F: definitely filled.
• X: definitely empty.
• ?: not yet decided.

An unknown cell is not an empty cell. That small distinction prevents a surprising amount of pencil-based confusion.

How does the overlap technique work?

Consider a separate five-cell line whose only clue is 3. Before any crossing information is known, its three possible completed arrangements are:

Position: 1 2 3 4 5
Option A: F F F X X
Option B: X F F F X
Option C: X X F F F

Position 3 is filled in all three options, so it is forced. Positions 1, 2, 4 and 5 still vary. The correct partial result is therefore ? ? F ? ?, not an entire block placed wherever it looks balanced.

This is overlap: find cells shared by every legal position of a block. Conceptis demonstrates the technique by moving a block to its extreme positions and filling their common area. Its step-by-step techniques guide also emphasizes recording known empty cells.

For a single block of four in five cells, the two placements occupy positions 1–4 or 2–5. Their shared positions are 2, 3 and 4. Notice that “a long clue” produces deductions, but not necessarily the full line at once.

How much space do several blocks need?

Add the block lengths and the mandatory gaps. A five-cell line with clue 2, 1 needs at least:

2 filled + 1 separating empty + 1 filled = 4 cells.

One extra cell remains, and it can occur before, between or after the blocks. Enumerating those possibilities gives:

F F X F X
F F X X F
X F F X F

The middle arrangement has two empty cells between blocks, which is allowed. “At least one” is not “exactly one.”

Conversely, clue 2, 2 fills a five-cell line completely once the required gap is included: F F X F F. Two blocks of two plus one gap use all five positions. Before shading a complicated line, count its required space; sometimes the arithmetic has already done the interesting part.

Try this original 5×5 nonogram

Rows are numbered top to bottom and columns left to right. The row clue is in the second column of the table. Each numbered column's clue appears in its heading. Commas separate distinct blocks.

For a text-only copy, the row clues are 1 / 3 / 5 / 1,1 / 1,1 and the column clues are 1 / 4 / 3 / 4 / 1. Every cell belongs to both a row and a column; both sets of requirements must hold.

Stop here if you want to solve it independently. The walkthrough follows immediately. You do not need to recognize a picture, assume symmetry, or add any rule not stated above.

Step 1: Fill the complete middle row

R3 has clue 5 in a line containing exactly five cells. Fill all of R3.

You now know R3C1, R3C2, R3C3, R3C4 and R3C5 are filled. Those coordinates name one cell each: R3C2 means row 3, column 2.

Do not mark a neighboring row merely because the emerging shape suggests it. The next deduction comes from the column clues.

Step 2: Use overlap in columns 2 and 4

Both C2 and C4 have clue 4 in five positions. Each four-cell block could start at R1 or R2. Either way, it includes R2, R3 and R4.

Fill R2C2 and R4C2, then R2C4 and R4C4. The R3 cells in those columns were already filled. Leave the top and bottom cells of these columns undecided for now.

At this point the new information is:

R1: ? ? ? ? ?
R2: ? F ? F ?
R3: F F F F F
R4: ? F ? F ?
R5: ? ? ? ? ?

We have learned something about four cells without choosing either possible starting position. That is deduction doing its job.

Step 3: Finish the outer columns

C1 and C5 each have clue 1. Their required single filled cell is already present in R3, so every other cell in those columns must be empty.

Mark R1, R2, R4 and R5 with X in both C1 and C5. This is not optional tidying: those empty marks constrain the remaining row possibilities.

If you want a broader method for recording deductions and their reasons, our systematic logic-puzzle guide uses the same distinction between a fact and an assumption.

Step 4: Complete the bottom two rows

R4 and R5 each have clue 1, 1, and both now have empty outer cells. Only C2, C3 and C4 remain available.

Two single filled cells with a gap fit into those three positions in exactly one way: F X F. Thus both complete rows are:

R4: X F X F X
R5: X F X F X

R4's filled cells confirm earlier overlap deductions. R5's filled cells are new. The X marks in C3 will now settle that column.

Step 5: Finish the center column and check the top

C3 has clue 3, but R4C3 and R5C3 are empty. Its consecutive block can only occupy R1, R2 and R3. Fill those three cells.

R1 now contains its one required filled cell at C3. Mark every other R1 cell empty. R2 has filled cells at C2, C3 and C4, with its outer cells already empty, so its clue 3 is satisfied.

The completed grid is:

    C1 C2 C3 C4 C5
R1: X  X  F  X  X
R2: X  F  F  F  X
R3: F  F  F  F  F
R4: X  F  X  F  X
R5: X  F  X  F  X

How do you know this solution is correct and unique?

Read the filled runs back from the grid. The rows give 1; 3; 5; 1,1; 1,1. The columns give 1; 4; 3; 4; 1. Every clue matches.

There are 13 filled cells: the row total is 1 + 3 + 5 + 2 + 2 = 13, and the column total is 1 + 4 + 3 + 4 + 1 = 13. Matching totals alone would not prove correctness; the run-by-run check does the necessary work.

We also exhaustively checked this original puzzle. The five rows individually allow 5, 3, 1, 6 and 6 arrangements. That gives 5 × 3 × 1 × 6 × 6 = 540 row-compatible grids. Exactly one also satisfies all five column clues: the grid shown above. The walkthrough explains it without branching; enumeration provides a separate uniqueness check.

For the next puzzle, repeat the same loop: inspect a line, make only forced marks, then revisit crossing lines. If nothing follows, leave the cell unknown. Forcing an answer is not a deduction, however confidently the pencil moves. More logic puzzles offer practice without turning speed into a score for the person solving them.

Sources

• Conceptis: Black-and-white and color nonogram rules.
• Conceptis: Nonogram solving techniques.

The 5×5 puzzle, line examples and calculations on this page are original worked examples, independently checked by exhaustive enumeration.