Reading a Square, Reading a Dot

Two arguments, both entirely local, both obvious once seen. Between them they finish every easy board on this site and they open every board at every level.

The square: a number against its own four sides

Take any numbered square and sort its four sides into three piles: drawn, crossed out, and not yet decided. If the drawn pile already matches the number, the square is finished and every undecided side of it can be crossed out at once. If the drawn pile plus the undecided pile only just reaches the number, there is no slack at all and every undecided side must be drawn.

A zero is the first of those in its purest and most valuable form. Nothing is drawn, nothing is owed, so all four sides go at once — four segments settled from a single glance at a single number, before anything else on the board has been considered. It is the cheapest information a slitherlink ever offers and it is why the opening move on every board is to find the zeros.

What happens next is where the famous pattern comes from. Those four crossed segments belong to the neighbouring squares as well, and a 3 sitting beside a 0 has just lost one of its four sides. Three remain and it needs three, so all three are drawn immediately. People learn that as a separate rule about a zero next to a three; it is this rule applied twice, which is a much more useful thing to know because it also applies to a 2 beside a 0, and to a 3 beside a segment crossed for any other reason at all.

Before — a 0 at the left of the middle row

After — all four of its sides are crossed out

The 0 sits at row 3, column 1. A zero says the loop uses none of that square’s four sides, so all four are crossed at once — the top, the bottom, the left along the board edge and the right shared with the 1 beside it. Nothing has been drawn, and four segments are nevertheless settled for good.

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The dot: no line, or exactly two

Every dot on a finished board has either nothing touching it or exactly two segments. Three would be a junction and four a crossing, and the loop is allowed neither. Applied while you work, that one sentence has three faces and a solver uses all of them without ever consciously separating them.

Two lines already meet at this dot, so it is full and everything else touching it is crossed out. Or one line arrives and every way out but one has been ruled out, so the line takes the remaining route. Those two are what most people mean when they say they are "following the line", and on a board that is moving they account for most of the moves.

The third face is the quiet one and it is the one that gets skipped. Nothing is drawn at this dot, and every segment touching it but one has been crossed off — so that last segment cannot be drawn either, because a line that reaches a dot and stops dead is not part of any loop. Nothing appears on the board when you apply it, which is precisely why people forget it, and what it does is take a side away from the two squares it touches.

Before — a 3 with one side already gone

After — its other three sides are drawn

The 3 at row 3, column 4 has a 0 immediately to its right, and the previous move crossed out that 0’s left side — which is the 3’s right side. Three sides are left and the 3 needs three of them, so all three are drawn. This is the pattern everybody learns as a rule about a zero beside a three, and it is the square rule used twice.

Why these two rules are all an easy board needs

An easy board on this site is one the solver finishes using nothing but these two arguments. That is the definition rather than an approximation of it: the solver takes the cheapest available step every time, and if it never had to reach past the second rung, the board is easy and is labelled so.

Which also means an easy board is guaranteed finishable by a reader who knows only what is on this page. If yours has stalled, the overwhelming likelihood is a dot beside something you have just settled where the line now has exactly one way out — the two rules feed each other constantly, and the whole skill at this level is remembering to go back to the dots after every number and back to the numbers after every dot.

Before — a line arrives at a dot with one way left

After — it can only go upwards

At the dot in row 3, column 5 of the lattice a line arrives from the left. Of the other three routes away from it, two have already been crossed out by the zeros on either side, so exactly one is left and the line has to take it. No number was consulted at all; the whole argument is that a line arriving somewhere has to leave again.

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 is the first thing to do on a slitherlink board?

Cross out all four sides of every 0. It is free, it settles four segments each time, and it very often forces a neighbouring 3 to draw its remaining three sides straight away.

Why cross out segments at all if nothing gets drawn?

Because a crossed segment is a result. It removes a route from two dots and a side from two squares, and almost every deduction above the first two rungs becomes visible only once eliminations have been recorded.

Can these two rules finish a whole board?

On an easy board here, yes — that is what easy is defined to mean on this site. From medium upwards they will stall somewhere and a corner or a pair of threes takes over.