How to Solve Nemologic, from Basics to Advanced Techniques

Nemologic is a logic puzzle where you work out which cells to paint using only the numbers attached to each row and column of a grid. Plenty of numbers are involved, but you never do arithmetic with them; the whole task is settling, one cell at a time, which cells must be painted and which can never be painted. Once the grid is complete, the painted cells form a picture, which is why it's also called a picture logic puzzle or nonogram.

The modern form is believed to have been devised independently by two people in Japan in the late 1980s, and it quickly found a home running in Japanese puzzle magazines. It picked up the English name nonogram after appearing in a British newspaper around 1990, and has since spread widely through newspapers, magazines, handheld games, and the web. Different media call it by different names; we use the name Nemologic, long used in Korea.

How to read the numbers

The numbers to the left of a row or above a column give, in order, the lengths of the consecutively painted blocks in that line. If a row reads 3 1, that row has a three-cell block first, followed by a one-cell block. The order is fixed left to right and top to bottom, so the one-cell block never comes before the three-cell one.

There is a gap between blocks

Whenever a line has two or more numbers, at least one empty cell separates each block from the next. Without that rule, there would be no way to distinguish 3 1 from 4, so this gap is really part of how the numbers count at all. Put the other way: if the sum of the numbers plus the required gaps between them equals the line's length exactly, that whole line is settled on the spot.

Zero and the finish condition

A line with a single 0 has nothing to paint, so you can leave the whole line empty. The puzzle is finished once every row and every column matches its own numbers. There is no time limit and no required order; you can work on any line first.

From here on, this section covers what this specific app does. Nemologic apps differ in these details, so reading this once should save you some confusion.

Five by five, one puzzle a day

The board is five cells wide and five tall, twenty-five cells total. It is generated from that day's date, so anyone who visits on the same day gets the same puzzle, and a new one opens when the date changes. Generating and solving both happen entirely in your browser, so nothing is exchanged with a server and no account is needed.

Only puzzles with exactly one solution

The generation order is a little unusual. We first draw a complete picture from that day's value, then derive the row and column numbers from it. The important part comes next: we count exactly how many pictures those numbers alone could produce. If the count is exactly one, that puzzle ships as is; if it's two or more, we discard the picture, change the seed, and draw again from scratch. Rather than carving cells out of a fixed picture to tune difficulty, we redraw the whole thing and filter.

Counting works like this. A five-cell line has only thirty-two possible layouts, so for each row we list every layout that matches its numbers, then lay rows down one at a time from the top, folding any branch the moment the columns built so far conflict with the column numbers. Counting stops the instant a second solution turns up, since at that point we only need to know there's more than one. That's why the puzzle you see on screen always has exactly one solution, and there is never a point where guessing is structurally required.

Each cell is drawn to be painted with slightly better than even odds, so painted cells generally outnumber empty ones, and an entirely empty picture is filtered out automatically. The value drawn is random, though, so you won't know what the finished picture is ahead of time. Guessing at the shape doesn't help in this app; reasoning from the numbers alone is always faster.

Two modes: paint and X

A button in the top right switches between paint mode and X mark mode. In paint mode, tapping an empty cell paints it, and tapping again clears it. X mark mode works the same way with an X instead. On a computer, right-clicking places an X regardless of the current mode. The two marks overwrite each other, so right-clicking a painted cell turns it straight into an X.

An X is just a note you leave yourself saying this cell is empty; it is never checked when judging completion. Only painted cells are compared against the answer, so you never have to place a single X, and leaving an answer-empty cell without an X never keeps the puzzle from counting as finished.

Wrong cells are never flagged

This is worth knowing up front. There is no feature that turns a wrongly painted cell red or raises a warning. The app checks the answer exactly once, when verifying that all twenty-five cells match the solution, so before that moment the board reacts the same way no matter what you paint. If the board looks full but the solved message doesn't appear, a cell is quietly wrong somewhere.

We also state clearly what's missing. Hints are free up to five times a day (see the three hints above), but there is no button that shows the whole answer, no undo, and no timer or mistake counter. Finished lines are not dimmed for you, and confirmed empty cells are not auto-marked with X. Every judgment call is yours. On the other hand, there's no limit on how many times you paint and clear, so a single mistake never ruins the puzzle.

Saved progress and finishing

Painted cells and X marks are saved in this browser, so they survive a refresh or closing and reopening the tab. An untouched board is not saved at all, and when the date changes, yesterday's save is discarded and a new puzzle opens. If a saved grid doesn't match the current cell count or its shape is broken, it is not loaded and you start from an empty board.

The moment your painted cells exactly match the answer, the solved message appears and the board locks. A share button then shows up, and the picture itself is only included in the share text once you've actually finished. If you press share while still mid-puzzle, it carries only the date and the fact that you're in progress, so sending it to someone else never leaks the answer.

These are things you work out by eye and by reasoning; the app won't find them for you. Once learned, though, they carry over to any size of board, not just this one.

Start where a whole line is forced

If the sum of a line's numbers plus the required gaps between them equals the line's length, that line has exactly one possible layout. On a five-cell line, that includes not just 5, but also 2 2, 1 3, and 1 1 1. With 2 2, there's no other way to fit two cells, a gap, and two more cells into five slots. Finding and filling these lines first, as soon as the puzzle opens, gives a head start to every other line that crosses them.

Always paint the overlap

Even a line that isn't fully forced can still yield settled cells. Slide a block to the start of the line and then to the end, and compare the two positions; any cell painted in both is painted no matter how the line is actually laid out. On a five-cell line, a 4 forces the middle three cells, and a 3 forces the middle one. Larger numbers produce wider overlaps, so start with whichever line in the whole board carries the biggest number.

Finish a line, then clear the rest

Once you've placed every block a line needs, every remaining cell in that line must be empty. If a 3 1 line has both its blocks placed, the one cell left over is obviously an X, no further thought needed. The app won't mark it for you, so do it yourself. These X marks narrow down where blocks can fit in the lines that cross them, so clearing is just as important as painting.

Past a certain point, the three techniques above stop producing new cells for a while. From here you need to move between lines instead of staring at just one, and the app still does none of this for you.

Anchor to an edge

Once you're sure the very last cell of a line is painted, the first block has nowhere to start but that end. If the number is 3, the three end cells are painted outright and the cell right after becomes an X. The reverse also helps: an X on the end cell effectively shortens the usable length of that line by one, which often exposes an overlap that wasn't visible a moment ago. Every time you settle a new cell, it's worth re-checking both ends of its line.

Cross over into the lines a painted cell touches

The heart of this puzzle is that every cell belongs to a row and a column at the same time. Once you settle a cell in a row, the column it passes through has just received new information worth a second look. Staring at the same stuck line rarely helps; switching to the line running the other way through a cell you just touched usually unlocks more.

Check your work by counting painted cells

The sum of all five row numbers and the sum of all five column numbers must be equal, since both describe the same total number of painted cells in the puzzle. Partway through, counting your currently painted cells against that total tells you how much is left. If you've already exceeded it, a cell is wrongly painted somewhere, and you can catch it yourself before waiting for the solved message that never comes.

Keep assumptions short

Because every puzzle has exactly one solution, it can in principle be solved without ever assuming anything. Still, when you're stuck, you can tentatively assume a cell is painted and follow that assumption through its row and column. A contradiction confirms that cell is empty. Since there's no undo, keep any assumption to two or three moves and only make one you can mentally unwind on your own.

Everything looks painted, but it's not marked solved

As noted above, this app never flags wrong cells. If there's no room left to paint but the solved message never appears, you either missed a cell that should be painted or painted one that shouldn't be. Start by comparing the sum of the five row numbers to your current painted count. If the numbers match but nothing happens, the cells themselves are in the wrong places, so it's faster to check each row one at a time against its actual blocks.

Only looking at single-number lines

A line with one number looks easy, but small numbers like 1 or 2 have a lot of room to move, so they rarely settle much. A number that fills most of the line, like 4 or 2 2, actually settles more cells. Prioritize lines with little spare room, not lines that merely look simple.

Skipping the X marks

If you only paint and never mark X, you end up recomputing each line's state in your head every time. On a board this small it's manageable, but leaving confirmed empty cells as X makes the remaining space obvious the next time you look at that line. X marks don't affect judging, so feel free to place and clear them freely.

Feeling the urge to guess

Because this puzzle has exactly one solution, there is never a point where guessing is required to move forward. If you're stuck, there's logic you haven't spotted yet, not a sign the puzzle has turned into a coin flip. On top of that, wrong cells are never flagged and there's no undo, so a single guess makes it hard to trace back where things went wrong. It's better to fill in only the cells you're confident about and take a short break.