Nonogram

Each row has a list of numbers beside it and each column has one above it. Those numbers are the lengths of the painted runs in that line, in the order they appear, and consecutive runs are separated by at least one blank square. A row marked `4 2` in a fifteen-wide grid holds a block of four, then a gap of at least one, then a block of two — somewhere. Your job is to work out exactly where, using the columns to narrow the rows and the rows to narrow the columns, until the painted squares form a picture.

The second mark matters as much as the first. A cross means this square is definitely blank, and placing crosses is not tidiness: it is what turns one deduction into the next. On this board a tap paints, the Cross out key swaps what a tap does, and on a mouse the right button crosses without changing anything.

The minimum column is the one that does the work. A line's numbers plus one gap between each pair is the least room they can occupy; the slack is the line length minus that total, and the slack is the only freedom the line has. A ten-wide line reading `4 2` has seven squares of content and three of slack, so no run can move more than three squares from its leftmost position. That single subtraction is where most deductions start.

Push a run as far left as it will go, then as far right, and paint anything covered both times. For a single run of length b in a line of length n that is exactly 2b − n squares — positive whenever the run is longer than half the line. A 4 in a row of 5 forces three squares. A 7 in a row of 10 forces four. A 12 in a row of 15 forces nine. Even an 8 in a row of 15 forces one, and that one square is often the whole opening.

Nothing else is known at that point, which is what makes it the right first move: it needs no crosses, no neighbouring line and no assumption. Sweep every row that way and then every column — the columns already have the rows' squares to work with — and the median 10 × 10 in this library comes out of that one pass with 78 of its 100 squares already settled.

Because it is checked, picture by picture, before it ships. Each of the 62 drawings in the library is turned into its row and column numbers and then handed to two independent routines: a line solver, which deduces only what every arrangement of a line agrees on, and a full search that counts solutions and stops at two. A picture ships only if the search finds exactly one solution and the line solver finishes the grid on its own. With the left-to-right mirror of each drawing that gives 91 distinct puzzles — 32 at 5 × 5, 34 at 10 × 10 and 25 at 15 × 15 — and every one of them has passed both tests.

This is not a formality. A grid of random squares passes the uniqueness test surprisingly often and the line-solvability test far less often, and a generated nonogram that quietly needs trial and error is the single commonest defect in the genre. It is also why the pictures are drawn by hand rather than generated: a random grid that happens to be solvable still looks like static when it is finished, and the finished picture is the entire reward.

Not here, ever. The gate every puzzle passes is stronger than uniqueness: it must be finishable by line logic alone, meaning a solver that only ever looks at one row or one column at a time, keeps every arrangement of that line still compatible with what is marked, and paints or crosses whatever all of them agree on. Applied over and over, that finishes every shipped puzzle: on the hardest 15 × 15 in the library, eight consecutive sweeps of all thirty lines each turn something new up, and only the ninth comes back empty.

So if you are stuck, there is a line somewhere that settles something, and the hint button will name it. It also checks your marks against the picture first, because one square painted in the wrong place makes every deduction after it sound and wrong at the same time.

The 5 × 5 is a two-minute puzzle and the place to learn what a cross is for. The 10 × 10 is the standard size of the genre and where most of the library sits. The 15 × 15 is where a picture can carry detail — between 48 and 173 of its 225 squares end up painted — and where a row of eight forces its middle before anything at all is known.