Runs of Signs, and Where They Pin

This one costs nothing and needs no numbers on the grid at all. Everything it has to say is available before you have made a single mark.

Counting the cells above and below

Take a cell and follow every sign leading away from it that points to something larger. Count how many distinct cells are forced to be above it that way, following the signs as far as they go. If two cells must be larger than this one, then this one cannot be n and cannot be n minus one either — it is at most n minus two. Now count downwards for the mirror image: three cells forced below means this cell is at least 4.

The famous case is a line with every sign pointing the same way, which settles the line outright: five cells ascending in a 5×5 can only be 1, 2, 3, 4, 5 in that order, and there is nothing left to work out. That case is rare and it is not what makes this technique worth its rung. The everyday case is much smaller and much more useful — a cell with one sign above it is not the largest value, a cell with two is not the second largest either, and on an empty grid that is dozens of possibilities gone in one pass.

The chains do not have to run along a line. A cell can have a cell above it to the right and another above it below, and both count towards the same total, because the rule only cares how many cells must hold a larger number than this one. Following the chains round a corner is where most of the yield is, and it is what people miss when they read the signs one row at a time.

Before — an untouched grid, and one cell with two cells below it
	1	2	3	4	5
1				∨
2	<		<
3			∨	>
4	1	5			3
5	3

After — it cannot be a 1 and it cannot be a 2
	1	2	3	4	5
1				∨
2	<		<
3			∨	>
4	1	5			3
5	3

The cell at row 1, column 4 is greater than the one below it, and that one is greater than its neighbour to the left. Two cells are therefore forced to hold smaller numbers than it, so it cannot be a 1 or a 2 and its candidate list drops to 3, 4 and 5. Nothing on the grid was read to reach that — only the two signs.

what the reasoning read what it decides what it wrote

Why this is the opening rather than a technique

Everything this argument can say depends only on where the signs were printed. It does not read a given number, it does not read a candidate list, and nothing you subsequently do to the grid can give it anything new to say. So it is worth doing once, thoroughly, at the very start, and then never again.

That is unusual and it is why this rung sits below the sweep rather than above it. Every other technique on this ladder becomes available and then becomes available again as the grid changes around it. This one is a single pass over the printed page. On an empty 9×9 with fifty signs it removes several hundred candidates before anybody has written anything, which is a strange and rather satisfying way to start a puzzle.

On paper the practical form is to write the range rather than the possibilities. A cell at the foot of a chain of two in a 6×6 is somewhere in 1 to 4, and writing "1-4" in the corner is faster than pencilling four digits and clearer to read back. Convert to real pencil marks once the sweep starts cutting into the middle of the ranges.

What it cannot do

It cannot use anything you have learned. A cell whose neighbour has just been narrowed to "3 or 4" is now constrained far more tightly than the chain count suggests, and this rung has no way to notice — it is still counting cells rather than reading candidates. That is a different technique with a page of its own, and it sits three rungs higher precisely because it has to be redone every time anything changes.

It also cannot see a contradiction between two chains. If the signs were printed so that a cell had to be larger than itself, the grid would have no answer at all, and that is a defect in the puzzle rather than a hard grid — the engine here checks for it and treats such a grid as broken rather than as a challenge.

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

What is a sign chain in futoshiki?

A run of inequality signs linking three or more cells so that each is larger than the last. The cells at the ends of the run are the ones it pins, and a run spanning a whole line settles that line outright.

Do sign chains have to stay in one row?

No, and the ones that turn a corner are the ones most often missed. The argument only counts how many cells must be larger or smaller than the one you are looking at, wherever they sit.

Is it worth redoing this technique later in a grid?

No. Everything it can say comes from where the signs are printed, which never changes, so one thorough pass at the start extracts all of it. Reading signs against live candidates later is a different and harder technique.