What a Corner Gives Away

Everywhere else a dot has four segments and has to wait. At the four corners of the board it has two, and the number sitting there is decided immediately.

Both or neither, and what that settles

The dot at the very corner of the board touches exactly two segments: one running along the top edge and one running down the side. The rule that a dot carries no line or exactly two therefore reduces to something much sharper here — both of those segments are drawn, or neither is. There is no third possibility and no waiting for the rest of the board to catch up.

Read that against the number in the corner square and two of the three cases collapse at once. A 3 in the corner cannot have neither, because it would then be left with two sides for a number that wants three, so both corner segments are drawn. A 1 in the corner cannot have both, because that is already two sides for a number that wants one, so both are crossed out. Either way, two segments are settled from a single number before a line exists anywhere on the board.

These are opening moves in the most literal sense: they need no state at all and can be applied to a completely blank board. That makes them a strange kind of technique — you never have to look for the right moment, only to remember that the four corners are there.

Before — a 3 in the top-left corner, nothing drawn

After — both of its outer sides are drawn

The corner dot at row 1, column 1 of the lattice touches only two segments: the top of the 3 and its left side. Both or neither, and neither would leave the 3 with two sides for a number that needs three — so both are drawn. Nothing else on the board has been read, and the first two segments of the loop are down.

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The 2 in the corner, which is the pretty one

A 2 in the corner survives both cases, and the argument does not stop there. Suppose the corner is used: the square has its two sides, so its other two are crossed, and the dot one step along the top edge has a line arriving from the corner with the route downwards already gone — it must continue along the edge. Now suppose the corner is not used: the square’s two far sides are drawn instead, and the dot one step along the top edge has a line arriving from below and again must continue along the edge.

Two entirely different positions, and the same conclusion in both. The second segment along the top edge is drawn either way, and by the identical argument turned ninety degrees, so is the second segment down the side. You have proved something about the loop without deciding anything about the corner itself, which is the first time in this puzzle that reasoning about a case you cannot resolve pays off.

It is worth doing that argument by hand once rather than memorising the picture. The same shape of reasoning is what the closing rule at the top of the ladder is made of, and a solver who has followed it here recognises it there.

Why the corners are exactly one step up

A medium board here is one the solver finishes with the two local rules and the corners, and nothing harder. It is a real threshold rather than a gradual one: an easy board never needs a corner to be read to keep going, and a medium board contains at least one moment where nothing at all happens until one is.

The moment is easy to recognise once you know to look for it. Every zero has been used, every dot with a line at it has been followed, every number has been counted against its sides, and the board is still nearly blank — because the deduction that unlocks it is sitting in a corner you have been looking past since you started.

On a very small board the corners occasionally cannot be the deciding factor at all, because five columns leave the two local rules enough to finish on. That is why asking for medium at 5×5 sometimes returns an easy board: the label is what the solver measured, not what was requested.

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

What does a 3 in the corner of a slitherlink tell you?

That both of its outer sides are on the loop. The corner dot offers only those two segments and must carry both or neither, and neither would leave the 3 one side short.

What about a 1 in the corner?

Both of its outer sides are crossed out, by the same argument run the other way: carrying both would give the square two sides when it only wants one.

Does the corner rule work anywhere except the four corners?

Not in this form. It depends on a dot having only two segments, which happens nowhere else — a dot along an edge has three and a dot in the middle has four.