Every nonogram opens the same way. Before the grid asks you to think, it hands out free cells — lines the clue arithmetic has already decided — and starting well is nothing more than collecting them in a fixed order: empty and full lines first, forced lines next, then the guaranteed middles of the fattest clues. Guessing never enters into it.
The numbers beside the grid are a complete description of the picture, compressed line by line. Solving is extraction: read a line, take what it proves, move on. What beginners usually lack is not technique but an order — which lines to read first, what each one owes you, and when to stop expecting gifts and start working the crossings.
Where do you start?
Start with the lines that have no freedom. A clue of 0 empties its whole line. A single clue equal to the grid’s width fills one. And a clue set whose runs plus their forced gaps exactly cover the width places itself completely. Harvest all of these before doing anything clever — they cost nothing.
The habit that makes this work: read every row clue and every column clue once, before making a single mark. You are not studying the average lines; you are scanning for extremes. A 0 here, a full-width run there, a crowded set like 6, 3 on a ten-wide row — those are the corners of the opening.
The measure of a crowded line is its wiggle room. Add the run lengths plus one forced gap between each pair; that sum is the minimum span the clues occupy. Subtract the span from the width, and what remains is the line’s entire freedom. Wiggle room of zero means the line has no choices left — it is already solved, waiting to be written in. Small wiggle room means the fat runs are pinned in the middle even while their ends still float, and pinned cells are exactly what the opening is for.
The first pass, worked in full
Here is the whole opening on one set of rows, each ten cells wide, read in the order the ritual asks.
- Clues: 0. The row stays empty. Mark all ten cells with an X and never read this row again. An X is a mark like any other — it feeds the columns exactly as filled cells do.
- Clues: 10. One run, the whole width. Fill all ten cells. Done.
- Clues: 6, 3. Span: 6 + 1 + 3 = 10. Wiggle room: zero. The row writes itself — six filled, one X, three filled — before you have read a single column.
- Clues: 7. Span 7, wiggle room 3. The leftmost placement covers cells 1 through 7; the rightmost covers 4 through 10. Every placement agrees on cells 4 through 7 — mark those four as filled. The rule in one line: a run keeps its middle, and the middle is run length minus wiggle room. Seven minus three, four guaranteed cells.
- Clues: 5, 2. Span: 5 + 1 + 2 = 8. Wiggle room: 2. The 5-run keeps 5 − 2 = 3 cells — cells 3 through 5, where its leftmost placement (1 through 5) and its rightmost (3 through 7) agree. The 2-run is not longer than the wiggle room, so it keeps nothing yet. That is not a failure; it is a note left for later.
Every one of these five rows gave up marks before a single column was read. Real openings are rarely that generous — plenty of lines give nothing at first — but the crowded ones pay immediately, and the empty ones pay in X’s, which are worth just as much.
What do you do when the free cells run out?
The grid stops giving and starts trading. Every cell you marked in a row also lives in a column, and it has changed that column’s arithmetic. From here on you alternate: read the columns your new marks landed in, take what they now prove, and follow those marks back into the rows they touch.
One concrete trade. Take the X that a forced row like 6, 3 leaves in its seventh column, and say it lands three cells from the top of a column ten tall whose only clue is 6. A run of six cannot fit in the two cells above the X, so it lives entirely in the seven below. A 6 inside a seven-cell stretch has wiggle room 1 — five cells, the fifth through the ninth, are guaranteed. One X bought five filled cells in a line you had not even started.
Sustaining that loop — line after line, marking only what cannot be wrong — is the whole middle of the puzzle, and it deserves a slower treatment than an opening guide can give it. The line-by-line method is that treatment: one line at a time, certainty accumulating, no guesses anywhere.
Where does it go wrong?
Three mistakes account for most ruined grids: trusting the emerging picture over the clues, skipping the X marks that record what you have proven, and estimating overlaps instead of counting them. Each one has a tell you can catch early — and each is cheaper to prevent than to unwind.
Solving the picture instead of the clues. The grid starts to look like a cat, so the ear gets filled in where an ear ought to go. The tell is a cell you cannot point to a clue for. The picture is the reward for the logic, never a source of it — if you cannot name the run a cell belongs to, you have not proven the cell.
Hoarding fills and skipping X’s. Filled cells feel like progress and X’s feel like bookkeeping, so beginners collect one and neglect the other. The tell is re-deriving the same line twice: you worked out yesterday that the run cannot reach those cells, and without the X the knowledge evaporated. X every cell the moment you prove it empty, and when a line’s clue is satisfied, X everything that remains in it at once.
Eyeballing the overlap. Somewhere in the middle of a long solve, the arithmetic gets estimated: that run looks like it reaches about there. The tell arrives three lines later, as a contradiction that costs a whole corner. Width, span, wiggle room, subtract — four small numbers, every time. The counting is the technique.
None of this asks for brilliance. The opening is arithmetic and order: take the empty and the full, take the forced lines, take the guaranteed middles, and then let the rows and columns answer each other until the picture arrives. Certainty first; speed follows on its own. The grids are patient, and they teach the patience back.
Questions
Do you solve the rows or the columns of a nonogram first?
Neither on principle. Open with whichever lines have extreme arithmetic — empty, full, or forced — wherever they sit, then alternate: each mark in a row changes what its column can hold, and the fastest deductions follow those changes.
Can you solve a nonogram without ever guessing?
Yes. A properly constructed nonogram has one solution reachable by deduction alone. Feeling forced to guess almost always means a deduction is waiting in the other direction, or an earlier mark is wrong.
What is the overlap rule in nonograms?
Subtract a line's minimum span — its run lengths plus one forced gap between each pair — from the line's width to get the wiggle room. Any run longer than the wiggle room is guaranteed to cover its middle, and the count of guaranteed cells is the run's length minus the wiggle room.
How do you know when a line is finished?
When the filled runs match the clue exactly — right lengths, right order. Mark every remaining cell in that line with an X at once; those X's are what the crossing lines read next.