How to solve a nonogram.

Each row and column carries a list of numbers. The numbers are the lengths of the filled runs in that line, in order, with at least one empty square between them. A row of ten squares clued 4 4 holds a run of four, a gap, and a run of four. Where those runs sit is the puzzle.

Two marks are worth making as you go. Shade a square you have proved filled, and put a light cross in a square you have proved empty. The crosses feel optional and are not: they are what stops you working the same square out twice.

Start by looking for clues that leave no choice. A ten-square row clued 10 has one arrangement, so the whole row is filled before you have looked at anything else. The same goes for any clue whose runs and gaps add up to the length of the line: 4 5 in a ten-square row needs four, a gap, and five, which is exactly ten.

These lines cost nothing and give the crossing lines something to work with, which is what the rest of the solve runs on.

This is the technique that opens most puzzles. Take a long run in a line with a little room to move. Push the run as far left as it goes, then as far right as it goes. Any square covered both times is filled, whichever position turns out to be right.

A run of seven in ten squares can start at square one, two, three or four. Slide it left and it covers squares one to seven; slide it right and it covers four to ten. Squares four to seven are covered by both, so they are filled, and you have four squares without knowing where the run begins.

The same works with several runs at once. Each run gets its own leftmost and rightmost position, and each overlap is filled.

The rule of thumb: a run overlaps itself when it is longer than the slack in the line. Slack is the length of the line minus the space the clue needs at minimum. A ten-square row clued 3 3 needs seven squares, so it has three of slack, and neither run of three beats it.

A line like that is not stuck, it is just waiting. One square from a crossing line is enough to break it open.

A single known square is worth more than it looks, especially at the end of a line. Take the row clued 3 3 that overlap could not touch. Learn that its first square is filled and the first run has nowhere else to go: it must run from square one to square three, and square four must be empty, because a fourth filled square would make the run too long.

That cross on square four is the useful part. It is a fact about the column that crosses there, and columns are where you go next.

Nothing above uses more than one line at a time. The solve comes from alternating: work the rows until they stop giving anything, then work the columns with what the rows just told you, then go back to the rows. Each pass makes the next one easier, and a puzzle that looked stuck two minutes ago falls over.

That loop is the whole method. Below, it solves a puzzle in three passes.

Some puzzles run out of line reasoning before they are finished. At that point you pick a square, assume it is filled, and follow the consequences until either the puzzle finishes or a line becomes impossible. An impossible line proves the assumption wrong, which makes the opposite true, and that is a deduction rather than a guess.

It only works when the puzzle has exactly one solution. Every puzzle in these books was machine-checked for that before it was printed, which is why a contradiction you find is always real information.

The easy volume never asks for it. Those puzzles come out under line reasoning alone, which is what makes them a good place to learn. The hard volume is built around the technique.