Nonogram solving basics

What the numbers actually say

Every row and every column carries a list of numbers. Each number is the length of one unbroken run of filled squares, the numbers are given in the order those runs appear, and every pair of runs is separated by at least one blank.

Two consequences are worth internalising early:

• A clue of 4 2 needs at least 4 + 1 + 2 = 7 squares. On a ten-square line that leaves only three squares of slack for the whole arrangement to slide about in.
• A clue of 0, or an empty clue, means the line is entirely blank. Cross the whole thing off at once — these are free and people routinely skip them.

Start every puzzle by finding the lines with the least slack. A line whose clues nearly fill it is where the forced squares are.

Overlap: the technique that starts every puzzle

Here is the one idea that turns a blank grid into a foothold. Take a line and push its runs as far left as they will go. Then push them as far right as they will go. Any square filled in both extremes is filled in every possible arrangement, so you can fill it in now.

The arithmetic is quick once you have done it a few times: a single run of length k on a line of length L gives you k − (L − k) certain squares in the middle. Any positive result is free progress. For multi-number clues the same idea applies to each run separately, using the leftmost and rightmost positions that run can occupy.

Sweep every row with this, then every column, before you attempt anything else. On a fifteen-wide grid the long clues give up a great deal immediately.

Marking blanks is half the puzzle

New solvers fill squares and leave everything else undecided. That is throwing away half your information. A square you have proved empty constrains the crossing line exactly as much as a filled one does — often more, because it chops a line into short segments that a long run cannot fit into.

Mark a blank whenever you can show it, in particular:

• Once a run is finished, the square at each end of it must be blank.
• Once a line's clues are all satisfied, every remaining square in it is blank.
• Any gap too short to hold the smallest remaining run is blank all the way along.

On our board you mark a blank by dragging with the right mouse button, or by holding Shift. It is worth building the habit even when it feels like extra work, because the crossing lines are where those marks pay off.

Working row against column

After the first overlap sweep, the puzzle becomes a loop. Fill and mark what a row forces, then look at the columns those squares sit in and see what has become forced there, then go back to the rows those columns touch. Progress moves like a crack spreading, not like a scan.

1. Follow the last thing you learned. After filling a square, immediately read its column. That is where the next deduction almost always is.
2. Re-read completed lines. When a line's runs are all placed, cross off the rest of it. Those crosses feed the crossing lines.
3. Look for the edges. A filled square at the very start of a line pins the first run exactly. Edges are much more informative than the middle.
4. Count what is left. If a line's clues total seven filled squares and you have six, the last one is somewhere specific — often findable by elimination.

The picture is a hint, and a trap

Halfway through, the image starts to become recognisable, and it is tempting to fill in what you expect to see. Resist it slightly. The picture is a good way to notice a square worth checking, and a bad way to justify filling it.

The safe version of the trick: when you think you can see what the drawing is, use it to choose which line to attack next, then prove that line from its clues. You get the speed benefit of the guess without the risk of an error you will not detect for another twenty squares.

When the last square goes in, the board tells you. Until then, the clues are the only authority.