The Assumption, and Why We Refuse It

It works on everything, which is exactly the problem. Here is what it is, how to do it properly if you meet one elsewhere, and why nothing on this site requires it.

How it works

Choose an empty cell. Write in one of the two symbols provisionally, and then follow the consequences with every ordinary technique — pairs, counting, line analysis, twin elimination — until either the grid finishes or something breaks: three alike in a line, a quota exceeded, two identical rows. If it breaks, the symbol you assumed was wrong, and the cell takes the other one. That much is a sound deduction and can be written in permanently.

The discipline is in the bookkeeping. Everything derived under an assumption is provisional and has to come out again if the assumption fails, so it needs to be visibly separated from what you knew before — a different colour on paper, or the undo button here, which is what it is for. A solver who cannot cleanly unwind an assumption will eventually contaminate a grid with a consequence of something that turned out to be false, and that is a far worse mess than being stuck.

Choose the cell carefully if you must do it at all. The best candidate is one in a line that is nearly full, because the consequences arrive within a move or two and the branch resolves quickly. Assuming into an empty region produces a long chain that neither confirms nor contradicts anything for ages.

Before — nothing ordinary applies
	1	2	3	4	5	6
1	1			1
2				0
3				0
4	0	1	0	1	1	0
5	1			0
6	0	0	1	1	0	1

After — one cell, proved by contradiction
	1	2	3	4	5	6
1	1	1		1
2				0
3				0
4	0	1	0	1	1	0
5	1			0
6	0	0	1	1	0	1

This position comes from a grid carved for uniqueness alone, with no check that a person could follow it — the kind this site refuses to publish. No pair, no gap, no completed quota and no twin will move it. Assuming a 0 in row 1 column 2 and following it breaks a rule, so the cell is a 1. Sound, and thoroughly unsatisfying.

the cell it decides what it wrote

Why it is not a difficulty level

Because it works on every grid without exception. Any puzzle with a unique solution can be finished by assuming, following, and unwinding — repeatedly if necessary — so "needs assumption" describes the solver’s patience rather than the puzzle’s structure. A scale whose top rung fits everything is not measuring anything.

Worse, it is what a generator reaches for when it has stopped checking. Carve clues out of a grid while only tracking that the solution stays unique, and you get puzzles that are unique and unsolvable by reasoning, which is the defect behind most "impossible" complaints about binary puzzle apps. The puzzle is not impossible. It is simply not a puzzle any more, and the app cannot tell the difference because it never looked.

So the generator here never removes a clue unless the deduction solver can still finish the grid without it. Anything that would have needed an assumption was rejected while it was being built, not graded afterwards. The position below is what one of those rejected grids looks like part-way through: everything ordinary has run out, and only a trial will move it.

If you meet one in the wild

Assume into the most constrained cell you can find, keep the provisional symbols visibly separate, and stop the moment you hit a contradiction rather than pressing on to see how bad it gets. One resolved cell is a full result — go back to the cheap techniques immediately, because that cell will usually have created several new pairs.

And if you find yourself nesting assumptions, put the puzzle down. Two levels of provisional working is where the errors come from, and a grid that needs it was generated by something that never checked whether a person could solve it.

Where this sits on the ladder

Rung 1Double
Two alike side by side push the opposite symbol out to both ends.
Rung 2Sandwich
Two alike with one gap between them: the gap takes the other symbol.
Rung 3Counting
A line already holding its full share of one symbol has none left to give.
Rung 4Line analysis
List every way one line can be finished and keep what they agree on.
Rung 5Twin elimination
A finishing that would copy a finished line is not a finishing at all.
Rung 6Assumption
Put a symbol in, follow it until it breaks a rule, and write the other one.

Common questions

Is trial and error allowed in a binary puzzle?

It is sound reasoning — an assumption that leads to a broken rule proves the opposite symbol. But it works on every grid, so a puzzle that requires it has not been checked for human solvability rather than being especially hard.

Do any puzzles on this site need an assumption?

No. A clue is only removed during generation while the deduction solver can still finish the grid without it, so every published puzzle is completable by ordinary reasoning.

Why do some binary puzzle apps produce unsolvable-feeling grids?

Because they carve clues while checking only that the solution stays unique. Unique and humanly solvable are different properties, and a generator that tests the first without the second will ship puzzles that can only be finished by trial.