How to Solve Picture Cross (Nonograms)

What a clue tells you

A row clue of "3 1" in a 10-wide row means: somewhere in the row there is a run of three filled squares, then at least one empty square, then a run of one filled square, and nothing else. The runs appear in that order from left to right (top to bottom for columns). The clue says nothing about where the runs sit, only their lengths and order.

Two numbers are worth checking first in any line: the total ink (3 + 1 = 4) and the minimum length the clue needs (3 + 1 + 1 for the gap = 5). The gap between the minimum length and the line's length is the slack. Low slack means you can place squares immediately.

Technique 1: overlap

The single most powerful idea. Imagine the clue's runs pushed as far left as they will go, then as far right. Any square that a run covers in both positions must be filled, because the run cannot avoid it.

For a clue of 7 in a 10-wide row, pushed left it covers squares 1 to 7, pushed right 4 to 10. Squares 4 to 7 are covered either way: fill them. For "3 1" in 10, the slack is 5, which is more than the 3-run's length, so nothing overlaps yet. The rule of thumb: a run overlaps with itself whenever its length is greater than the slack, and the number of squares you can fill is the length minus the slack.

Technique 2: edges and zeros

• A clue of 0 means the whole line is empty: mark every square with ×.
• A clue equal to the line's length fills the whole line.
• If the first square of a line is filled, the first run starts there: fill it to its full length and mark the square after it empty.
• If a filled square sits within a run's length of the edge and the first clue is that run, the squares between the edge and the filled square may also be forced (a run of 4 with a filled square in position 2 must include positions 1 or 3; if position 1 is known empty, positions 2 to 5 are filled).

Technique 3: mark the empties

Beginners only fill squares. Experts mark empty squares with × just as eagerly, because an empty square splits a line into segments, and a run must fit inside a single segment. A run of 5 cannot fit in a segment of 4, so that whole segment is empty. When every run in a clue has been placed, every other square in the line is empty; mark them, because those × marks are information for the crossing lines.

The Picture Cross board lets you drag to fill or mark several squares at once; use it to mark whole empty segments in one stroke.

Technique 4: which run is this?

Once some squares are filled, ask which clue each filled square belongs to. If a filled square is too far from the left edge to be part of the first run (because the first run plus its gap could not reach it), it belongs to a later run. Conversely, a filled square near the left edge must belong to the first run. Attaching squares to runs narrows where the other runs can go, which creates new overlaps.

If a filled square could belong to two different runs, leave it and look elsewhere; the crossing line often decides.

Technique 5: completed runs

When a run of filled squares reaches the length of the largest clue in that line, it is complete and must have an empty square at each end. If a run of filled squares is longer than any single clue, you have made a mistake. When a run reaches the length of a clue and the clues on either side are different lengths, you can often say exactly which clue it is.

Working the whole grid

1. Do every row and column once with overlap, edges and zeros. Big clues first: a clue that nearly fills its line gives the most.
2. Mark the empties implied by every placement.
3. Return to lines where new squares (filled or empty) appeared; recompute the overlap within the segment that is left.
4. Use "which run is this?" on lines with several runs.
5. Alternate rows and columns. Lines you cannot progress on now will move once their crossings change.

Never guess. If a grid seems stuck, check the lines with the least slack again; there is always a line that moves.

Common mistakes

• Forgetting the gaps. Runs need at least one empty square between them; the minimum length includes the gaps.
• Filling squares that merely could be filled. Only fill what every arrangement of the clue requires.
• Not marking empties. Half of the deductions come from × marks.
• Reading clues in the wrong order. Row clues read left to right, column clues top to bottom.
• Trusting the picture. On hard days the picture is not obvious until late; do not fill a square because it "looks like it should be part of the ear".

Glossary

• Run (or block): a group of consecutive filled squares.
• Clue: the list of run lengths for a line, in order.
• Slack: the line length minus the minimum length the clue needs.
• Overlap: the squares a run covers whether pushed left or right.
• Segment: a stretch of a line between known empty squares.