Slitherlink: 0s, 3s and corners
A few small patterns start almost every Slitherlink. Each one is a short proof: try the other way, and a dot or a number breaks.
How it works
Most Slitherlink puzzles open with the same few patterns. You can learn them as pictures, but each one is a tiny proof by contradiction: suppose a side went the other way, and some dot or number would break.
The rule behind all of them: every dot has two lines or none. A dot in the corner of the grid has only two sides, so they are both lines or both ×.
Why the pair works
Suppose the middle side were ×. Each 3 would need its other three sides, and together they make a ring around the two squares. That ring is the whole loop, so every other number on the board goes unmet. So the middle side is a line.
Now suppose the left 3 skipped its far side. It would use its top and bottom. The dot at the top of the middle line would be full, so the right 3 couldn’t use its top and would take its bottom. Then the dot at the bottom of the middle line has three lines. So the far sides are lines too.
The sides that would carry the middle line straight on, above and below the pair, are ×. The middle line turns at both ends.
Worth memorizing
A 1 in a corner crosses both corner sides.
A 3 in a corner draws both corner sides.
A 3 beside a 0 draws its other three sides.
Two 3s side by side draw three parallel lines.
The middle line turns at both ends.
Practice
0 of 4 rightWhich sides of this corner 3 must be lines?
Which sides of this corner 1 can you cross?
How many sides of this 3 are forced to be lines?
Two 3s sit side by side on the top row. How many lines are forced right away?