A nonogram that forces a guess is a broken nonogram. The 20 by 20 trophy board in Nonogram is not one of those: every cell can be derived, one line at a time, from the clues and from cells you have already proved. This page states what that guarantee means, shows the measurements behind it, and gives you a way to confirm it at the board.
Two guarantees, and why you need both
People mean one of two different things by "this puzzle is fair", and they are not the same claim.
The first is uniqueness: exactly one grid satisfies the clues. This board has that, confirmed by an exhaustive search that found one solution and then went looking for a second without finding it. Uniqueness alone is weak comfort, though. A puzzle can have exactly one answer and still be unreachable without trial and error, if getting there means assuming a cell, following the consequences and backing out when they collapse.
The second is line solvability: the answer can be reached by reading one row or one column at a time, never holding two hypotheses at once. This board has that too, and it is the guarantee that matters while you play. Fixing only the cells every valid arrangement of a line agrees on, the grid closes completely: 296 of 400 cells on the first sweep, 388 after the second, all 400 after the third. A fourth sweep finds nothing left to do, which is how you know the process ended rather than stopped.
What "never stuck" means in numbers
Solving the board greedily, always taking whichever single line pays the most at that moment, the grid closes in 41 steps. The interesting figure is not 41. It is that no step in the sequence returned zero. At every position along the way, including the last four steps where only one cell remained to be proved, at least one line still had a forced cell in it. There is no point on this board where every line shrugs and the only move left is to pick a cell and hope.
| Measurement | Result |
| Cells settled by pure line reading | 400 of 400 (100.00%) |
| Sweeps needed | 3, with a 4th confirming nothing is left |
| Greedy line reads to finish | 41 (21 rows, 20 columns) |
| Steps that produced nothing | 0 |
| Distinct solutions | 1 |
| Painted cells in the answer | 210 of 400 (52.50%) |
Why a guess costs more here than you think
There is no undo, no mistake counter and no score on this board. That sounds forgiving, and in one sense it is: a wrong cell costs you nothing but the time to erase it. But it also means a wrong guess is silent. The game says nothing when you paint a cell that does not belong to the answer. It simply withholds the finish, and it withholds it in exactly the same way whether you are one cell wrong or forty.
The check is a set comparison: the painted cells must be precisely the 210 of the answer, and nothing partial registers. So a wrong guess announces itself much later, as an unexplained failure to finish, by which time you have built deductions on top of it. That is the practical argument for never guessing here: the feedback arrives too late to be useful.
Worked Example: the two moments that feel like a guess
The lone 2 at the edge. Column 1 carries a single clue of 2. In a column of twenty that block has 19 possible positions and no cell is painted in all of them, so a cold read gives nothing. This is where players start guessing, because a clue that yields nothing feels like a clue that needs help. It does not; it needs four cells from elsewhere. Once rows 1 and 20 are known empty from their clue of 0, and rows 5 and 6 known painted from their clue of 3 12 3, one of the 19 positions survives and column 1 hands over 16 cells at once, all empty. The line that looked like it needed a guess was being read too early.
The three scattered blocks. Columns 4 and 17 carry a clue of 2 2 2, which on a blank column allows 455 different arrangements, more than any other line on the board. Two lines out of forty account for 910 of the 2,640 arrangements the whole blank grid admits. Nothing about those columns can be settled by looking at them, and looking harder does not help. What settles them is arriving late: by the time the rows and the central columns have done their work, almost every cell in columns 4 and 17 is already known, and the arrangements that still agree with the board have collapsed to a handful. Both columns then finish in a single read.
Both moments share a shape: the line is not ambiguous, it is under-informed. Leave it and read a line that just changed.
The stall test
When you feel stuck, run these three checks in order. On this board one of them always fires.
- Is there a line whose clue exactly fills its length? Add the clue numbers and add one for each gap between blocks. If the total is 20, that line has a single arrangement and is finished on sight. Rows 5 and 6 both qualify, and each is worth 20 cells.
- Is there a line with a clue of 0? The whole line is empty. Rows 1 and 20 qualify, and they are worth 20 cells each with no reasoning at all.
- Which line changed most recently, and which lines cross it? Every cell you fix updates twenty crossing lines. Read those, not the line you were staring at. Seventeen of the forty lines return nothing on a cold read, so sweeping in fixed order wastes much of the opening.
If all three come up empty, you have made an error somewhere, not found a position that needs a guess. The arithmetic that catches it is simple: the row clues sum to 210 and the column clues sum to 210. Those are the same 210 cells counted twice, once along each axis. Count the painted cells in any row you have finished and check it against its clue.
Recognising the picture is not solving it
The subtlest way to guess here is to stop deducing and start drawing. By the point where 176 of the 400 cells are settled, the trophy is unmistakable: the rim, the spine and the outer edges of both handles are on the grid, and filling in the rest by eye is tempting.
That is a guess wearing a disguise, and the board is built to punish it quietly. The stepped base is the place it goes wrong: rows 15, 16 and 17 widen by 6, 8 and 12, which is not the smooth taper the eye predicts from the rows above. Filling those rows by intuition produces a shape that looks right and fails the check. The deduction, by contrast, is trivial once you take it in order, because each of those rows is pinned by the central columns and settles in one read.
How to prove it to yourself at the board
You do not need to trust any of the figures above. Solve a sitting under one restriction: never fix a cell unless you can name the single line that forced it and say why. Not "it must be part of the handle" but "column 1 has one block of two, rows 5 and 6 are painted, so everything else in this column is empty."
If that restriction ever left you with no legal move, the board would require a guess. Carried to the end it never does, and the grid completes. The measurements on this page are the same walk done by machine, which is why they agree.
Frequently asked questions
Could this board have more than one valid picture?
No. An exhaustive search that kept going after finding the first solution, specifically to look for a second, found none. One grid satisfies the clues.
If I am stuck, does that mean the puzzle needs a guess?
It means an unread line is waiting. Every position reachable by correct deduction on this board has at least one line with a forced cell in it, measured across all 41 steps of a full solve. The line you are stuck on is usually not the line that can move.
Should I mark empty cells with X even though the game does not check them?
Marking them costs nothing and it is what makes the next deduction collapse. Column 1 pays out 16 cells only because four cells around it were already settled, two of them empty. You can hold that in your head, but not for long on this grid.
What is the safest opening if I do not want to think hard?
Rows 1 and 20 first, since a clue of 0 empties the whole line, then rows 5 and 6, whose clue fits their row exactly. That is 80 cells settled in four reads without a single cross-reference, and it immediately unlocks the four central columns.
Does painting the whole grid win by brute force?
No. The answer has 210 painted cells and a fully painted grid has 400. The check compares the painted set against the answer, so extra cells do not count as containing the solution.
What happens if I erase a cell after solving it?
The finish steps back until you restore it. The check runs on the current state of the board, so the grid has to be exactly right at the moment you look at it.
Related guides
- How to Play Nonogram for the rules and the standard techniques in general form.
- The Nonogram line solving method for the deduction this page depends on, with its measured yield.
- The trophy walkthrough if you would rather follow the whole solve in order.
- How to Solve Sokoban Puzzles for the same no-guess discipline on a very different board.
- How to Play Minesweeper and Puzzle games beginners guide for where to take the habit next.