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Puzzle Quarry

Cells That Own a Set of Values

The top of the published ladder, and the only rung that reasons about several cells at once rather than about one cell and its neighbours.

The naked set: two cells, two values, no idea which way round

Find two cells in the same row whose candidate lists are both exactly "3 or 4". Between them they are going to use up the 3 and the 4 — one each, in some order nobody yet knows — so no other cell in that row can hold either value, and both can be struck out everywhere else along the line. The deduction is about the pair jointly; asking which of them is the 3 is a question you do not need to answer and probably cannot.

The same argument runs for three cells whose candidates between them come to exactly three values, which is where it starts to be genuinely hard to spot by eye. The three lists need not be identical: cells reading "1 or 5", "1 or 2" and "2 or 5" between them own 1, 2 and 5, and that is a valid triple even though no two of them match. Nearly everybody who learns pairs misses triples for a long time for exactly this reason.

What makes it expert is not the reasoning, which is simple enough to state in one sentence, but the search. You are looking for a coincidence between candidate lists across a line, and there is no single cell you can look at that announces it. The habit that finds them is to scan a line for the cells with the shortest lists and compare those to each other first.

Before — two cells in the row can only be a 3 or a 4
1234
1∧3
2∧
3∨∧
4>
After — both values leave the rest of the row
1234
1∧3
2∧
3∨∧
4>
In row 3, the first cell and the third are both down to a 3 or a 4. Between them they will use both values up, whichever way round they turn out to be, so neither value can appear anywhere else in that row — and the second cell, which had all four candidates, is cut in half at a stroke.

what the reasoning read what it decides what it wrote

The hidden set: two values, two cells

The mirror image, and nobody learns one of these without the other. Take a row and ask where the 2 could go, then where the 6 could go. If both questions give the same two cells, then those two cells hold the 2 and the 6 between them — so everything else in either of their candidate lists can be struck out, however long those lists were.

It reads the opposite way round from the naked set and it fires in situations the naked set cannot touch, because it works on cells whose candidate lists are still long. Two cells reading "1, 2, 4, 6" and "2, 3, 6" look like nothing at all until you notice that the 2 and the 6 have nowhere else in the row to be, at which point both cells collapse to "2 or 6" and the line usually falls apart shortly afterwards.

In practice it comes up less often than the naked pair and more often than the naked triple, and it is by a distance the most satisfying move in the puzzle — a great deal of the grid disappears at once, from an observation about two numbers that were not obviously connected to each other.

The move that is not on this ladder

There is a deduction that ought to belong here and does not, and it is worth explaining because it looks so promising. Two cells either side of a sign, both down to exactly the same two candidates, are settled outright: the sign says which is the larger, so the smaller value goes to the point and the larger to the open end. It is sound and it is elegant.

It is also unreachable. Reading a sign against its neighbour’s live candidates gets there first, every single time — the moment the larger cell’s ceiling is known, the smaller cell loses the top value by ordinary sign bounds and the pair is broken up before this argument can be applied to it. A finder for it was written into the engine and run across every size and every level, and it never once fired.

So it was taken out. A rung that cannot fire is not a harmless extra: it is a difficulty label that lies, because a grid could be described as needing a technique that no solver would ever have to use. The same standard applies to every rung on this ladder, and it is why there are five rather than six.

Where this sits on the ladder

Common questions

What is a naked pair in futoshiki?

Two cells in the same row or column whose candidate lists are the same two values. They will use both values between them, so those values can be struck out of every other cell in that line.

Do the cells of a naked triple have to have identical candidates?

No, and that is why triples are missed so often. Three cells reading "1 or 5", "1 or 2" and "2 or 5" own exactly three values between them and count as a triple.

How do I find these without checking every combination?

Scan a line for the cells with the shortest candidate lists and compare those to each other first. A pair is a coincidence between two short lists, and long lists almost never take part in one.

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