The Rules of Slitherlink

Three rules, and the third one is the puzzle. Here is each of them shown on a board small enough to take in at a glance rather than merely stated.

Rule one: a number counts its own four sides

Every square on the board has four sides — the four segments joining the dots at its corners. A number written in a square says exactly how many of those four the finished loop runs along. Not at least, not at most, and not a count of anything else: not neighbours, not corners, not the squares around it.

That is worth labouring because it is the mistake beginners make and it makes half a board look impossible. A 1 does not mean the loop passes near the square once. It means precisely one of the four segments bounding it is drawn and the other three are not. A 0 means all four are unused, which is the single most useful thing that can be written on a board — one number and four segments settled.

Nothing says a square must carry a number. Most do not, and an unnumbered square is not a hidden zero: the loop may use none, one, two, three or all four of its sides, and which of those it does is decided by everything except that square.

Nine squares, nine numbers, and no line yet

The only loop those numbers allow

Rule two: never a junction, never a crossing

Look at any dot on the board when you have finished. Either no segment touches it at all, or exactly two do. Three would be a place where the line splits and you could not say which way it went; four would be a place where two lines cross. Both are forbidden, and stating the rule at the dots covers both cases with one sentence.

It is also the most useful form of the rule while you are working, because it is what makes a partly drawn line predictable. A line arriving at a dot has to leave it again, and if only one way out has not been crossed off, that is the way it goes. This is the argument that does most of the work on an ordinary board and it costs nothing to apply.

The mirror image is quieter and gets skipped. If nothing is drawn at a dot and every route away from it but one has been ruled out, that last one cannot be drawn either — a line that reaches a dot and stops dead is not part of any loop. Nothing appears on the board when you use this, which is exactly why people forget it exists.

Illegal — three lines meet at one dot

Legal — every dot carries none or two

Rule three: one loop, not two

When the board is finished, everything you have drawn has to be a single closed circuit. You must be able to set off along it from anywhere and come back to where you started, having walked the whole thing. Two neat separate rings, each perfectly closed, each obeying every number it touches, is not an answer and never becomes one.

Nothing about any individual number or any individual dot will tell you that you have broken this rule, which is precisely what makes it the interesting one. A board can satisfy every number on it and be plainly, permanently wrong. The figure below is exactly that: every number correct, and two rings where there should be one.

The rule is far more useful applied backwards than checked at the end. Instead of finishing and inspecting, ask of a segment you are considering whether drawing it would close a ring right now — and if it would, whether that ring could possibly be the whole answer. It usually could not, and that is a deduction available long before the board is anywhere near done.

Illegal — every number correct, and two separate rings

Legal — the same numbers, joined into one loop

Four things the rules are assumed to say and do not

They do not say the loop must visit every dot. Most boards leave a good many dots untouched, and a dot with nothing at it is the ordinary case rather than an unfinished one. Trying to sweep the line past everything produces a shape no set of numbers will ever agree with.

They do not say the loop must be convex, or smooth, or anywhere near symmetrical. Long thin spurs two squares wide are perfectly legal and turn up constantly, and a finished loop that looks like a coastline is not a sign that something has gone wrong.

They say nothing about diagonals. Segments run between horizontally or vertically neighbouring dots and nowhere else, so a loop can only ever turn at right angles.

And they do not permit guessing. Every board published here was checked to be finishable by argument alone, so a board that will not move is one where something has been missed rather than one that wants a coin flipped. That is a promise about the generator, and it is why the generator throws away most of what it builds.

Common questions

Can a slitherlink loop cross itself?

No. Every dot carries either no segments or exactly two, so four segments meeting at a dot — a crossing — is impossible, and so is three, which would be a branch.

What does a 0 mean in slitherlink?

That none of that square’s four sides are used by the loop. It is the most productive clue on any board, because a single 0 settles four segments before you have thought about anything else.

Is a board with every number satisfied always solved?

No, and that is the point of the third rule. Two separate closed rings can satisfy every number on the board and are still wrong; the finished drawing has to be one single loop.