Asking Where a Value Can Live

The same information, read the other way round. It is the move that separates players who finish medium grids from players who stall on them.

Turning the question round

Everything up to here looks at a cell and asks what it could be. Take one row, pick one number, and ask the opposite question: which cells in this row could still hold it? If the answer is a single cell, that cell holds it — and it holds it whatever else its own candidate list still contains, because the row has to contain that number somewhere and there is nowhere else left.

The reason this feels like a different technique rather than the same one is that the cell it settles usually looks entirely unremarkable. Its candidate list might still be three or four long. Nothing about the cell says "me"; the fact only appears when you look along the whole line for one value, and it stays invisible for as long as you keep asking cells about themselves.

Do it value by value rather than cell by cell. Take the 1 and walk the row, then the 2, then the 3. Trying to hold every value in your head at once is what makes people describe this as tiring, and it is unnecessary — one number and one line at a time is the whole discipline.

Before — three cells in the row, and only one of them can be a 4
	1	2	3	4
1		1	4
2	1		∧
3
4				4

After — the 4 takes its only remaining home
	1	2	3	4
1		1	4
2	1	4	∧
3
4				4

Row 2 opens with a given 1, and the two cells at its right-hand end are both down to a 2 or a 3. That leaves exactly one cell in the row that could still hold the 4, so it holds it — even though on its own that cell still looked like a 2, a 3 or a 4 and nothing about it stood out.

what the reasoning read what it decides what it wrote

Where it bites, and where it does not

It bites hardest on the largest and smallest values, because those are the ones the signs have already restricted. A 1 cannot sit at the open end of any sign and an n cannot sit at the point of one, so on a grid with plenty of signs the extreme values often have very few homes left in a line — and very few becomes one sooner than you expect.

It bites hardest on columns for the same reason it bites hardest on the values nobody is watching: attention drifts to rows because rows are how we read. A grid that will not move after every row has been questioned is very often one column away, and that is worth doing before reaching for anything harder.

It has nothing to say about a line where every remaining value still has three or four possible homes, which is most lines early on. That is why it is not the opening move: on an untouched grid it fires almost nowhere, and it becomes powerful only once the sweep has thinned the candidate lists enough for a value to run out of room.

What medium means, precisely

A medium grid here is one the solver finishes with the sign chains, the sweep and this question, and nothing harder. It is a real threshold rather than a gradual one: an easy grid never needs it at all, and a medium grid contains at least one moment where nothing whatsoever will happen until somebody asks a line where a particular value could go.

That moment is usually recognisable once you know to look for it. Every cell has two or three candidates, no cell is down to one, no sign is telling you anything new, and the grid feels like it has run out — which is exactly the shape of a grid with a value that has quietly run out of homes.

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 the hidden single technique in futoshiki?

Picking one value, scanning a row or column for the cells that could still hold it, and finding exactly one. That cell takes the value regardless of what else its own candidate list contains.

Why do I miss these when I can see naked singles easily?

Because the cell it settles looks ordinary — its candidate list is often still three long. The fact only shows up when you scan a whole line for one value rather than looking at cells one at a time.

Which values are worth checking first?

The largest and the smallest. The signs restrict them more than anything in the middle of the range, so they run out of homes in a line sooner than the others do.