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Puzzle Quarry

Filling On Spec, and Why We Refuse It

Sound reasoning, and useless as a measure of anything, because it cracks every board ever printed. Here is how to do it, and why no board here will ask.

Supposing a cell, and watching a line fall over

Pick an undecided cell and provisionally fill it. Follow the consequences with every ordinary argument — its row, then the columns that row touches, then the rows those columns touch — until either the board finishes or some line becomes impossible: a run that cannot fit, a number with nowhere left to go, a filled cell no run can reach. If a line breaks, that cell cannot be filled, and crossing it off is a sound permanent deduction.

The discipline is entirely in the bookkeeping. Everything derived under a trial is provisional and has to come off again if the trial fails, so it must be visibly different from what you knew before — a second colour on paper, or the undo button here, which is exactly what it is for. A solver who cannot cleanly unwind a trial will eventually leave a consequence of something false on the board, which is far worse than being stuck.

Choose the cell carefully if you are going to do it at all. The best candidate sits in a line with very little slack, because the consequences then arrive within a move or two and the branch resolves quickly. Trying a cell in a wide-open line produces a long chain that neither confirms nor contradicts anything for ages.

Before — nothing ordinary applies
11112211
2
11
21
2
1
After — one cell, decided by contradiction
11112211
2
11
21
2
1
This board has exactly one picture and no reasoned route to it — the kind this site grows, grades and then discards. Six ordinary moves have been made and every argument above has run out: no number left to overlap, no line settled at its ends, no pocket to eliminate, no block to glue and no line whose arrangements agree about anything. Supposing one cell filled breaks a line, so it is blank. Sound, and thoroughly unsatisfying.

what it decides what it wrote

A method that fits everything measures nothing

Because it works on everything without exception. Any board with a single answer can be finished by filling something, following it and unwinding, repeatedly if necessary, so "needs a guess" is a statement about the solver’s patience rather than about the board. A scale whose top rung fits every puzzle is not measuring anything at all.

It is also the difference this puzzle makes easiest to overlook. A nonogram with exactly one answer and a nonogram a person can reason through are genuinely different properties, and the gap between them is wide — a great many patterns have precisely one solution that no amount of line-by-line argument will ever reach. A generator that checks only for uniqueness will therefore ship those, and it will not know it has.

So the generator here asks both questions. Every board is put through the technique solver as well as the counter, and any that needed a guess is discarded and another grown in its place. The hand-drawn boards were put through the same check one at a time, and the ones that failed were redrawn. The position below is what one of the rejected boards looks like part-way through.

If a board from somewhere else forces your hand

Trial into the tightest line you can find, keep the provisional marks visibly separate, and stop the instant a line breaks rather than pressing on to see how bad it gets. One decided cell is a complete result — go straight back to the cheap arguments, because that cell will usually shorten a line or supply a wall and set off a run of ordinary moves.

The moment you catch yourself starting a second trial inside the first one, stop and put the board away. Nobody unwinds two layers of provisional filling reliably, and a board asking for it was made by something that never once checked whether a person could get through it.

Where this sits on the ladder

Common questions

Is trial and error allowed in a nonogram?

It is sound reasoning — a supposition that makes a line impossible proves the opposite. But it works on every board, so a board requiring it has not been checked for human solvability rather than being especially hard.

Will any nonogram here ever make me guess?

No. The generator discards every board that needs one, and the hand-drawn boards were checked individually; four had to be redrawn and one had to be abandoned entirely.

Why do some nonogram apps produce boards that feel impossible?

Because they check only that the answer is unique. On this puzzle the gap between unique and reachable by argument is unusually wide, so a generator testing the first without the second ships boards only a trial can finish.

More Nonogram pages