Guessing, and Why We Throw It Out

It finishes every grid ever made, which is precisely the problem. Here is how to do it properly if you meet one elsewhere, and why nothing here will make you.

What a trial looks like on a grid of numbers

Pick a cell with two candidates and provisionally write one of them in. Follow the consequences with every ordinary technique — sweep it out of its lines, reread the signs around it, look for values running out of homes — until either the grid finishes or something breaks: a cell with no candidates left, a value with nowhere to go in some line, a sign with no room on either side of it. If it breaks, that value was wrong, and striking it out is a sound permanent deduction.

All the discipline is in the bookkeeping. Everything derived under a trial is provisional and has to come off again if the trial fails, so it has to be visibly distinguishable 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 grid, and that is a much worse position to be in than being stuck.

Choose the cell carefully if you are going to do it at all. The best candidate is a cell in a line that is nearly full, because the consequences arrive within a move or two and the branch resolves quickly. Trying a value in a cell whose whole row is still wide open produces a long chain that neither confirms nor contradicts anything for ages.

Before — nothing ordinary applies
	1	2	3	4	5
1			<	<	∧
2		∧	∨
3		<
4			>	>
5			<

After — one candidate, struck out by contradiction
	1	2	3	4	5
1			<	<	∧
2		∧	∨
3		<
4			>	>
5			<

This grid has exactly one answer and no reasoned route to it — the kind this site carves, grades and then discards. Fifteen ordinary moves have been made and nothing above is available: no settled number left to sweep, no value down to one home, no sign that says anything new, no pair worth arguing about. Supposing the top-left corner is a 1 breaks the grid, so it is not. Sound, and thoroughly unsatisfying.

what it decides what it wrote

Why guessing cannot be a level of anything

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

It is also what a generator falls back on when it has stopped checking. Take a Latin square, thin the signs until only one arrangement survives, and you have a grid that is unique and unreachable by reasoning — which is the defect behind almost every "impossible" review a futoshiki app collects. The grid is not impossible. It has simply stopped being a puzzle, and the app cannot tell, because it never asked.

So the generator here asks. Every grid is put through the technique solver as well as the counter, and any that needed a guess is discarded and another carved in its place. The position below is what one of those rejected grids looks like part-way through: fifteen ordinary moves have been made, everything ordinary has run out, and only a trial will move it.

If a grid elsewhere forces one on you

Trial into the most constrained cell you can find, keep the provisional numbers visibly separate, and stop the instant you hit a contradiction rather than pressing on to see how bad it gets. One eliminated candidate is a complete result — go straight back to the cheap techniques, because that elimination will usually have settled a cell and set off a cascade.

And if you find yourself running a trial inside a trial, put the grid down. Two levels of provisional working is where the errors come from, and a grid that requires it was produced by something that never checked whether a person could follow it.

Where this sits on the ladder

Rung 1Sign chains
A run of signs pins its ends: below k cells you cannot exceed n minus k.
Rung 2Singles
A number that is settled is gone from the rest of its row and its column.
Rung 3Only one home
A value that fits just one cell of a row or column belongs in that cell.
Rung 4Sign bounds
Read a sign against what its neighbour can still hold, not against the whole range.
Rung 5Pairs and sets
Two cells that own two values between them lock those values away from the rest.
Rung 6Assumption
Write a number in, follow it until the grid breaks, and rub it out again.

Common questions

Is trial and error allowed in futoshiki?

It is sound reasoning — a supposition leading to a broken rule proves the opposite. But it works on every grid, so a grid requiring it has not been checked for human solvability rather than being especially hard.

Do any grids on this site need a guess?

No. Every grid is graded by the technique solver before it is published, and any that reached the trial rung was discarded and replaced rather than sold as a harder level.

Why do some futoshiki apps produce grids that feel impossible?

Because they check only that the answer is unique. Unique and humanly reachable are different properties, and a generator testing the first without the second will ship grids that only a trial can finish.