Pixel Logic

How to play

Every row has a list of numbers to its left, and every column has a list above it. Those numbers are the lengths of the runs of filled squares in that line, given in the order they appear, with at least one empty square between one run and the next. A row labelled 4 2 on a ten-wide grid contains a block of four filled squares, then a gap of one or more empty squares, then a block of two, and nothing else. A row labelled 0 is completely empty. The lists say nothing about where the blocks sit — that is the entire puzzle, and it is settled by comparing what a row allows with what its columns allow.

Filling a square says it belongs to the picture. Marking a square with an X says it does not. The X is only a note to yourself: the grid never checks your marks and you can finish a picture without placing a single one. What the grid does check is the filled squares, and they have to match the hidden picture exactly — every square of it filled, and no extra squares filled anywhere else.

Three numbers sit above the grid. Time starts on your first action rather than when the page loads, so opening the puzzle and wandering off costs nothing. Left is how many squares of the picture you have yet to fill in. Best is your quickest completion at that grid size, kept in your browser only. Ten pictures are included, four on the five by five grid and six on the ten by ten, and each one is named the moment you complete it.

Controls

• Left click or tap fills a square, and doing it again empties it.
• Right click places or removes an X.
• Press and hold on a touch screen places or removes an X, for the same reason.
• The mode button switches what a plain tap does, which is the reliable option on a phone where holding your finger down can be interpreted as a text selection.
• Arrow keys move between squares, Enter or Space fills the one you are on, and X marks it empty.
• Check reports how many of your filled squares do not belong to the picture, without naming them. Clear the grid wipes your work but keeps the same picture. New picture deals a different one at the current size.

Overlap, the move that opens almost every grid

The first squares you can be certain about usually come from a single line, with no cross-referencing at all. Take a row of ten squares with the clue 8. Push that block of eight as far left as it will go and it covers columns one to eight. Push it as far right and it covers columns three to ten. Whatever the truth is, it lies somewhere between those two extremes, so any square covered by both — columns three to eight — must be filled. That is six squares of certainty from one glance at one clue.

The arithmetic behind it is worth carrying around. Add the numbers in the clue, plus one gap between each pair, and subtract the total from the length of the line. Call the difference the slack: it is how far the whole arrangement can slide. Any block longer than the slack has a guaranteed core, and the core is the block length minus the slack. In the example above the slack was two, so the block of eight had a core of six.

A row of ten with the clue 4 3 needs four squares, one gap and three squares, which is eight, leaving a slack of two. The block of four gives you two certain squares and the block of three gives you one. That is only three squares out of ten, but three squares in a row become clues for three columns, and those columns hand you squares in other rows. This is the rhythm of the whole puzzle: a small certainty in one direction unlocks a small certainty in the other.

Why the X marks earn their keep

Beginners fill squares, ignore the marks, and stall halfway through a ten by ten. An empty square you have not marked still looks like a possibility, and possibilities are exactly what the overlap rule needs you to eliminate. Marking is how you shrink a line.

The most direct use is closing a finished block. If a row is labelled 3 and you have three filled squares in a row, the squares on either side of them must be empty, and marking them may immediately shorten the columns they belong to. The second use is measuring what is left. Once a line is broken up by X marks, it becomes a set of separate segments, and each remaining block has to fit inside one of them. A segment of two cannot hold a block of three, so the whole segment can be crossed out. That single observation resolves more ten by ten grids than any clever technique.

The third use is edges. If the leftmost square of a row is filled and the clue begins with 4, then the first four squares are filled and the fifth is empty, with no ambiguity at all. Edges are anchors, and marking the square that closes a block against an edge is often the move that starts a chain reaction across four or five columns.

When a line stops giving you anything, leave it and look at the line crossing it that has the largest numbers, since large numbers have the least room to move. Every picture here was checked with a solver that does nothing but this — one line at a time, filling in only what is true of every remaining arrangement for that line, going round and round the rows and columns until nothing changes. All ten grids finish that way, which is our guarantee that you never need to guess, and it also means each picture is the only answer that matches its clues.

Where the puzzle came from

Grid puzzles of this kind took shape in Japan in the late 1980s, out of competitions in which entrants drew a picture on squared paper and worked out the numbers that would describe it. Japanese puzzle magazines carried them shortly afterwards, and they reached newspapers in Britain in the early 1990s, which is when the name nonogram came into common use in English. Other names for the same idea have circulated ever since, usually attached to whichever publication or program a player met first.

The ten pictures on this page were drawn here, square by square, and each was adjusted until it solved cleanly by line logic — a picture that looks charming but leaves two possible answers is not a puzzle, it is a coin toss with extra steps.

Questions players ask