The Ring That Must Not Close

The one-loop rule looks like something you check at the end. Turned round it is the sharpest deduction in the puzzle, and it is what expert means here.

A closed ring is final, and that is the whole argument

Find two loose ends of the same piece of line — the two dots where a run of drawn segments stops — and notice that a single undecided segment joins them. Drawing it would close that run into a ring. And a closed ring can never grow again, not one segment, ever, because every dot on it already has its two lines and there is no room for a third.

So the question becomes very simple: could that ring be the entire answer? It could not if there is line drawn anywhere else on the board, because a second piece could never join it. It could not if any number on the board is left unsatisfied by it, because that number will never get another chance. Either of those makes the segment impossible, and it is crossed out.

What is being used is the connectedness rule read backwards. Rather than finishing the board and checking that it came out in one piece, you ask of each candidate segment whether drawing it would commit you to a piece that cannot be the whole thing. On a board two-thirds done there are usually several such segments, and each one is a free elimination.

Before — three sides of one square, and a gap

After — the gap is crossed out

Three segments already run around the unnumbered square at row 2, column 1, leaving two loose ends one segment apart at the dots in row 3, columns 1 and 2. Joining them would close a ring of four, and a closed ring can never grow — so it would have to be the whole answer, with the 3 in the top-right corner and everything else left unsatisfied. It cannot be, so the gap is crossed out.

what the reasoning read what it decides what it wrote

Where it turns up, and why it is worth waiting for

Almost never at the start, because it has nothing to work with: a board with three segments drawn has no runs long enough to nearly close. It becomes available in the middle game, when the line has grown into a few long meandering pieces, and it is at its most powerful late, when those pieces are groping towards each other and there are several places they could plausibly join.

The characteristic shape of an expert board is exactly that. It moves along under the cheap rules like anything else, arrives two-thirds finished with nothing forced anywhere, and then turns on a single observation about which of two nearly-touching ends may not be joined. Solvers who never learn this argument describe those boards as needing a guess. They do not; they need this.

The smallest case is worth carrying in your head because it is so common: three segments forming three sides of a single square, with the fourth undecided. Drawing that fourth closes a ring of four, which can only be the answer on a board where every number outside it is a zero. On any real board it is crossed out on sight.

Why nothing here is graded above this

Above it there is only the trial — draw something, follow it until the board breaks, rub it out — and that will finish any board at all, which is exactly why it cannot be a difficulty level. A scale whose top rung fits everything measures nothing.

So expert here means precisely this argument and no more: somewhere on the board there is a segment that nothing else will decide, and the reason it is decided is that using it would close a ring too soon. Boards that need anything beyond that are discarded during generation rather than sold as a harder level.

It is also the only rung on this ladder that looks at the whole board rather than at one square, one dot or one pair of numbers, which is a fair description of why it feels harder than everything below it even though the argument itself is a single sentence.

Where this sits on the ladder

Rung 1Counting a cell
A number with its sides already drawn has none left to give; a zero gives none at all.
Rung 2Following the line
Every dot carries no line or exactly two, so a line that arrives has to leave again.
Rung 3The corner cases
A corner dot has only two segments to offer, which settles a 1, a 2 or a 3 sitting in it.
Rung 4Threes together
Two threes side by side, or corner to corner, decide their outer sides before anything is drawn.
Rung 5No early closing
A segment that would seal a short ring with numbers still unsatisfied can never be drawn.
Rung 6Assumption
Draw a segment, follow it until the board breaks, and rub it out again.

Common questions

When can I not close a loop in slitherlink?

Whenever the ring you would close is not the entire answer — when there is line drawn outside it, or when any number on the board would be left unsatisfied by it. A closed ring can never grow, so either case makes the segment impossible.

Why does this only start mattering late in a board?

Because it reasons about runs of line, and early on there are none long enough to nearly close. It becomes available once the drawing has grown into a few long pieces with ends groping towards each other.

Is there a quick version I can spot on sight?

Three sides of a single square with the fourth undecided. Closing it makes a ring of four, which on any real board leaves numbers unsatisfied, so the fourth side is crossed out immediately.