How to solve Slitherlink: one loop through the dots

The rules

• Draw one closed loop along the dotted grid lines.
• A number says how many of its cell's four sides are part of the loop (0 to 3).
• The loop never crosses itself or branches: every dot has either no lines or exactly two.
• Cells without a number can have any number of loop sides.

The dot rule

This is the engine of every deduction: at each dot, the loop either passes through (two lines) or doesn't touch it (no lines). So:

• A dot with one line and only one free direction left must continue that way.
• A dot with two lines is finished: ✕ its other edges.
• A dot with three of its edges ✕-ed can't have a line on the fourth.

Patterns to learn

• 0: none of its four sides are used: ✕ them all.
• 3 in a corner: the two sides on the grid's edge are lines, since a corner dot can't be passed by a single line.
• Two 3s side by side: the line between them is used, and so are their outer sides parallel to it; the short edges continuing that middle line are ✕.
• Two 3s diagonally: each one's two outer corner sides (away from the other) are lines.
• A 0 next to a 3: the 3 uses its three sides away from the 0.
• 1 in a corner: its two edge sides are ✕, because using either would force the other.

Keep it one loop

Separate loops are not allowed. When two loose ends could join and close a small loop while other lines are still unconnected elsewhere, that edge must be ✕. Near the end, this rule often decides the last few edges.

A related trick is inside and outside. Every cell is either inside the loop or outside it, and everything beyond the grid's border is outside. Two neighbouring cells are on different sides exactly when the loop runs between them. So once you know which side two neighbouring cells are on, you know whether the edge between them is a line.

A solving routine

1. ✕ around every 0 and apply the corner and adjacent-3 patterns.
2. Go round every dot: finish dots with two lines, extend dots with one line and one way out.
3. Check every number: when its sides are decided, fill or ✕ the rest.
4. Avoid closing small loops early, and think inside and outside in stuck areas.

Practise

On Puzzimo, tap a line to draw it, tap again to mark it ✕ and a third time to clear it, or drag through the dots to draw several at once. Undo is free. Grids come in 5×5, 7×7 and 10×10, with a 5×5 daily challenge, and every puzzle has exactly one loop.

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