Picross — Play a Generated Nonogram
Play nonograms that always have one answer and never need a guess — plus the measurement showing sparse pictures do not make puzzles at all.
Click a square to fill it, again to mark it as definitely empty, again to clear. Every puzzle here has exactly one answer and can be finished by reasoning one line at a time — no guessing is ever required, and that is checked before the puzzle is handed to you.
The hard part is not the logic — it is getting one answer at all
The usual worry about a generated nonogram is whether it can be reasoned out or whether you will have to guess. Measured, that worry is misplaced. The thing that actually goes wrong is that a random picture usually has siblings — other pictures with exactly the same clues — and then there is no right answer to find. How often a random picture gives a puzzle with a single answer, by how full the grid is:
| Grid | 20% full | 30% full | 40% full | 50% full | 60% full | 70% full | 80% full |
|---|---|---|---|---|---|---|---|
| 5 × 5 | 34% | 41% | 54% | 72% | 87% | 95% | 99% |
| 10 × 10 | 0% | 2% | 16% | 53% | 83% | 96% | 100% |
| 15 × 15 | 0% | 0% | 3% | 37% | 75% | 94% | 100% |
A 15 × 15 grid with a fifth of its squares filled — which is what a simple drawing on a mostly empty grid looks like — produced a puzzle with a single answer zero times in two hundred attempts. Not rarely. Never. At 30% full, still never.
Meanwhile the guessing problem barely exists. Of the pictures that do give a single answer, here is the share you can finish by line-by-line reasoning with no guessing at all:
| Grid | 20% full | 30% full | 40% full | 50% full | 60% full | 70% full | 80% full |
|---|---|---|---|---|---|---|---|
| 5 × 5 | 98% | 96% | 98% | 99% | 99% | 100% | 100% |
| 10 × 10 | — | 57% | 74% | 93% | 99% | 100% | 100% |
| 15 × 15 | — | — | 33% | 92% | 99% | 100% | 100% |
Once a grid is more than half full it is essentially always 93% or better, and usually 100%. So a generator does not really need to worry about fairness — it needs to worry about ambiguity, and the two are solved by the same thing: filling more squares.
Which is why generated nonograms look like dense blobs rather than recognisable pictures. It is not a failure of artistry in the generator. Sparse pictures are simply not puzzles.
And the worst case is exactly n factorial
Enumerating every one of the 65,536 four-by-four grids: they give 58,196 distinct clue sets, and 52,362 of the grids (79.9%) have a clue set to themselves. The most overloaded clue set is shared by 24 different pictures.
Those 24 are not a curiosity — they are exactly the grids with one filled square in every row and every column. Their clues are all 1s in both directions, so every rearrangement reads identically and nothing can tell them apart. The count is the number of ways to arrange them, which is n factorial:
| Grid | Pictures sharing the all-1s clue set |
|---|---|
| 2 × 2 | 2 |
| 3 × 3 | 6 |
| 4 × 4 | 24 |
| 5 × 5 | 120 |
| 6 × 6 | 720 |
| 7 × 7 | 5,040 |
| 8 × 8 | 40,320 |
| 9 × 9 | 362,880 |
| 10 × 10 | 3,628,800 |
This is the mechanism behind the table above. A lone filled square with no neighbours can trade places with any other lone square in a different row and column, and the clues never notice. A sparse grid is mostly lone squares, so a sparse grid is mostly interchangeable — which is why the uniqueness column collapses to zero rather than merely getting worse.
How to use
- Pick a grid size and take the puzzle you are given.
- Click a square to fill it, again to mark it as definitely empty, again to clear.
- The numbers are the runs of filled squares in that line, in order.
- Every puzzle is checked to have one answer reachable without guessing.
Frequently asked questions
Do these puzzles ever need a guess?
No. Each one is accepted only if a solver that reasons a single line at a time can finish it. That solver never guesses, so a grid it completes is both the only answer and reachable by pure deduction.
What actually goes wrong when a nonogram is badly made?
Ambiguity, far more often than difficulty. A random picture usually shares its clues with other pictures, and then there is no right answer to find. Measured over random grids, a 15 by 15 with a fifth of its squares filled gave a single answer zero times in two hundred attempts.
Is it hard to make a nonogram that needs no guessing?
Much easier than it sounds. Among pictures that have a single answer at all, 93 per cent of 10 by 10 grids at half density can be finished by line-by-line reasoning, rising to 100 per cent once the grid is 70 per cent full. The guessing problem is real but small; the ambiguity problem is the one that bites.
Why do generated nonograms look like blobs rather than pictures?
Because sparse pictures are not puzzles. Uniqueness depends almost entirely on how full the grid is — zero per cent of sparse 15 by 15 grids work, against 100 per cent of grids that are 80 per cent full. A generator that insists on a single answer is forced into dense grids, and dense grids do not look like drawings.
What is the most ambiguous nonogram possible?
One filled square in every row and every column. All the clues are 1s in both directions, so every rearrangement reads exactly the same. The number of pictures sharing that clue set is n factorial — 24 at 4 by 4, and 3,628,800 at 10 by 10.
Why is a lone filled square a problem?
Because it can trade places with any other lone square in a different row and column without changing a single clue. A sparse grid is mostly lone squares, which is why sparse grids fail so completely rather than just getting a bit worse.
How do the clues work?
Each number is a run of consecutive filled squares in that line, read left to right or top to bottom, with at least one empty square between runs. A line with no numbers is completely empty.
Does this send anything anywhere?
No. The puzzle is generated and checked entirely in your browser.
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