A Sign, Read Against What Is Left

The opening pass read every sign against the numbers 1 to n. This reads them again, against what the cell on the other side can still actually be.

The two questions

Take a sign and look at the cell on its larger side. What is the biggest number that cell can still hold? Whatever it is, the cell on the smaller side must be below it — so if the larger side has come down to a 4 at most, the smaller side loses the 4, the 5 and everything above. Then run it the other way: look at the smallest number the smaller side can still hold, and the larger side loses that value and everything under it.

That is a strictly stronger reading than the opening pass, and the difference is where the technique lives. On an untouched grid the larger side could be n, so the smaller side merely loses n, which the chain count already told you. The moment anything at all narrows the larger side, the sign has something new to say — and it will have something new to say again the next time that cell narrows.

So a sign is not a clue you spend once. It is a relationship that keeps producing, and the practical habit is to revisit every sign touching any cell you have just changed. That is a small local sweep and it is the cheapest way to get value out of this rung without walking the whole grid.

Before — the cell on the large side of the sign has settled at 2
	1	2	3	4
1		3
2
3			∧
4			2>

After — its neighbour can only be a 1
	1	2	3	4
1		3
2
3			∧
4			2>	1

The sign in the bottom row says the cell at row 4, column 3 is greater than the one to its right, and that cell has just settled at 2. Its neighbour therefore has to be below 2, which strikes the 3 out of a candidate list reading 1 or 3 and leaves a single value. The opening pass could not have found this: it only knew the neighbour was not a 4.

what the reasoning read what it decides what it wrote

Why it sits above the cheaper moves

Because it subsumes them and costs more to perform. The opening chain count is this argument in the special case where nothing has been narrowed yet; the run of signs that settles a whole line is this argument applied n minus one times in a row. They come first because they are the versions you can do without consulting anything, and this is what you do when the same reasoning still applies but the numbers no longer announce themselves.

The cost is real. Every sign is two comparisons and two eliminations, there can be well over a hundred signs on a 9×9, and none of them stays answered — a sign you cleared five moves ago may be productive again now. Doing this exhaustively before the free sweeps have run dry is the single best way to turn this puzzle into arithmetic homework, which is the complaint people who dislike futoshiki almost always describe.

Done at the right moment it is remarkably sharp. Late in a grid the cells either side of a sign are frequently down to two candidates each, and at that point a single sign resolves both of them at once — the smaller value goes to the point of the sign and the larger to the open end, with no further thought required.

What makes a grid hard, precisely

A hard grid here is one where the solver, always taking the cheapest available move, was at some point forced into this reading — no settled number left to sweep, no value down to one home in any line, and the opening chain count long since exhausted. That is a statement about the grid containing a genuine bottleneck rather than a statement about its size.

It is also why size and difficulty are separate choices on this site. A 9×9 easy grid is eighty-one cells and no bottleneck at all; a 5×5 hard grid is twenty-five cells and at least one. Most players find the second takes longer, and nearly all of them find it more interesting.

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

How is this different from reading the sign chains?

The chain count uses only where the signs are printed and is done once. This reads each sign against the candidates its neighbour currently has, so it produces something new every time either cell narrows.

When should I start reading signs this way?

After the sweep and the where-can-this-value-live scan have both come up empty. Both are cheaper, and both shorten the candidate lists this technique has to work with.

Which signs are worth rechecking?

The ones touching whatever you changed most recently. A sign only has something new to say when one of its two cells has narrowed, so anywhere else is a wasted look.