Nonogram puzzles guide
Advanced Nonogram Techniques: Edges, Joining, Splitting and What-If
Stuck after the overlap method? Learn edge forcing, joining and splitting, capping blocks, clue assignment, gap fitting and contradiction, with worked examples.
When the overlap method stops giving you squares, the puzzle is not stuck — you are just out of easy moves. The techniques below all come from one idea: for each clue, track the full range of squares it could still occupy, and update that range every time a square is filled or crossed out. Edges, Xs and existing filled squares all shrink those ranges, and a shrunken range produces new squares and new Xs.
Every example uses a 10-square line. # is filled, x is empty, . is unknown, and the digits number the squares (0 means 10). Each "after" line shows everything that can be deduced from that line alone — we checked every one by listing all valid arrangements in a short script. If you need a refresher on the basics, read how to solve nonograms first.
Think in ranges, not positions
For each block, ask two questions: what is the leftmost square it could start on, and what is the rightmost square it could end on? Anything inside that window might belong to it; anything outside belongs to a different block or to nobody.
Clue 2 2, with squares 2 and 9 already filled:
1234567890 before .#......#. after .#.xxxx.#.
The first filled square (2) must belong to the first block, so that block lies within squares 1–3. The last filled square (9) must belong to the last block, so it lies within 8–10. Squares 4 to 7 are out of reach of both blocks, so they are empty. The overlap method sees nothing here; range thinking finds four Xs.
Habit to build: whenever a line changes, recompute the earliest start and latest end of its first and last blocks. Most advanced deductions are this, done carefully.
Edges and walls: forcing from a border
An edge of the grid and an X act the same way: a wall that a block cannot cross. A filled square close to a wall pushes its block away from the wall — sometimes called gluing, because the block is stuck to that square.
A filled square on the edge
Clue 3 2, square 1 filled. The first block must start on square 1, so squares 1–3 are filled and square 4 is empty:
1234567890 before #......... after ###x......
A filled square next to a wall
Clue 4 2, square 1 crossed out, square 2 filled. Square 1 now acts as the edge, so the 4-block starts exactly on square 2:
1234567890 before x#........ after x####x....
A filled square near a wall
Same clue without the X. The 4-block must include square 2, so it starts on square 1 or 2. Either way, squares 2–4 are filled:
1234567890 before .#........ after .###......
Rule of thumb: count squares from the wall. If the first block from that wall has length n and a filled square sits at position p with p no greater than n, every square from p to n is filled. Here p = 2 and n = 4, so squares 2–4.
Joining and splitting fragments
When a line contains two filled fragments with unknowns between them, decide whether they belong to the same block.
Joining
Clue 5, squares 3 and 6 filled. There is only one block, so both squares are part of it: fill 3 to 6. The block is now 4 long and needs one more square, on square 2 or square 7. Anything further away is out of reach:
1234567890 before ..#..#.... after x.####.xxx
Splitting
Clue 2 2, squares 4 and 6 filled, square 5 unknown. If square 5 were filled, squares 4–6 would form a block of 3, longer than any clue. So square 5 must be empty — the fragments belong to different blocks. That settles the whole line: the first 2 must include square 4 but cannot touch square 5, so it is 3–4; the second must include 6 and cannot use 5, so it is 6–7:
1234567890 before ...#.#.... after xx##x##xxx
The general test: if joining two fragments would create a block longer than the largest clue that could reach them, they must be split, and the square between them is empty.
Capping a finished block
When a run of filled squares reaches the length of the block it belongs to, put an X on both ends. The tricky part is knowing which clue the run is. Length is often enough.
Clue 1 3 1, squares 5–7 filled. A run of 3 can only be the 3-block — the 1s are too short. So square 4 and square 8 are empty:
1234567890 before ....###... after ...x###x..
This is one of the most used moves on big puzzles: as soon as a run equals the largest clue in its line, it is complete and can be capped, no matter where it is.
Working out which clue a square belongs to
Many stuck positions break open once you decide which clue owns a filled square. Clue 3 2, squares 6 and 9 filled:
1234567890 before .....#..#. after xxx.##..#.
- Could 6 and 9 be the same block? That block would span at least 6–9, which is 4 squares. The largest clue is 3, so no.
- So square 6 is the 3-block and square 9 is the 2-block. The 2 is the last clue, and nothing could own square 9 if square 6 were already the 2.
- Where can the 3 sit? It must include square 6, so it starts on 4, 5 or 6. Starting on 6 would cover 6–8 and touch the filled square 9, merging into a block of 4. So it starts on 4 or 5, and squares 5 and 6 are filled in both cases.
- Nothing comes before the 3, so squares 1–3 are empty.
Fitting clues into gaps
Xs split a line into separate gaps. Check which clues physically fit in each gap — and remember that clues after a block must fit too.
Clue 4 3, square 5 crossed out. The line is now a 4-square gap (1–4) and a 5-square gap (6–10). Could the 4-block go in the right-hand gap? Then the 3 must follow it in the same gap, needing 4 + 1 + 3 = 8 squares in a space of 5. Impossible. So the 4 fills squares 1–4 exactly, and the 3 sits in squares 6–10, where it has a slack of 2 and gives its middle square:
1234567890 before ....x..... after ####x..#..
Without that X, the same clue only gives squares 3, 4 and 8. One crossed-out square tripled the information.
The same check eliminates gaps that are too small for any remaining clue: a 2-square gap in a line whose unplaced clues are all 3 or more is entirely empty.
Contradiction: the what-if technique
When every line-level technique is exhausted, you can test a hypothesis. This is not guessing — it is proof by contradiction:
- Pick a square with only two outcomes that matter. The best candidates sit in a line with very few possible arrangements, or where filling the square would force a long chain of consequences.
- Assume one outcome — say, filled — and follow the consequences with normal logic in the crossing lines. On paper, use a pencil or a different mark so you can roll back.
- If you reach a contradiction — a line whose clue can no longer fit — the assumption was wrong. Undo everything from the trial and mark the opposite outcome. That square is now proved.
- If no contradiction appears, you have proved nothing. Undo the trial completely. The opposite assumption might still be the wrong one, and a trial that "seems to work" is not evidence. Try the other outcome or a different square.
Within a single line, you already used this idea in the clue-assignment example: "what if squares 6 and 9 were one block?" led straight to a contradiction. Across lines, chains get longer, so keep trials short: if a trial runs more than a handful of steps without a conflict, abandon it and look for a more constrained square.
Two-line contradiction tip: the most productive trials are squares where one outcome would complete a block (forcing Xs on both ends) in a line whose crossing lines are almost full. Completed blocks spread information quickly, so contradictions show up within two or three steps.
Which technique to reach for first
| You see… | Try… |
|---|---|
| A filled square within a block's length of an edge or X | Edge forcing |
| Two fragments with unknowns between them | Joining or splitting |
| A run as long as the largest clue | Capping |
| Filled squares and several clues, no obvious owner | Clue assignment with ranges |
| Xs dividing a line into pieces | Gap fitting |
| No line changes on a full sweep | A short what-if on the most constrained square |
In NonoPop, every puzzle is checked by a solver that reasons about one row or column at a time, and a puzzle is only accepted if that solver can finish it. So you never need a cross-line what-if there — a deduction from a single line, using the techniques above, is always available. It can still be quicker to test a hypothesis, and Undo makes rolling back easy. The same skills carry over to bigger grids, which we cover in our 25×25 strategy guide, and to color nonograms, where blocks of different colors can touch.
Frequently asked questions
Is using contradiction the same as guessing?
No. A guess commits to an answer and hopes. A what-if follows the logic until it breaks, and only the outcome that was proved impossible is used. If the trial does not break, you undo it and learn nothing — you do not keep it.
What is a line solver?
A method — usually a computer program — that looks at one line, considers every arrangement of its blocks that matches the squares already known, and marks any square that is the same in all of them. Every technique in this guide is a human shortcut for part of that idea.
Why do I keep getting stuck on hard puzzles?
Usually a missed X. Re-check lines where blocks are complete but not capped, and lines with gaps too small for any remaining clue. Empty squares are easy to overlook and often unlock the next step.