A cell already has its number of fence segments (rule out the rest), or needs every undecided edge to reach it (draw them all).
Dot rulesT2
The loop passes through each dot 0 or 2 times. A dot with two lines rules out its other edges; a dot with one line and one open edge must continue; a lone open edge at an empty dot is ruled out.
Fence parityT3
The fence separates inside from outside. A fence edge with a known side puts the other side opposite; an × joins like sides; two known neighbours force the edge between them (fence if they differ, × if they match). The world beyond the border is always outside, so a border fence puts its cell inside.
Corner reasoningT4
Short case analysis at a cell's corner. A 3 whose corner is sealed from outside must use both corner edges; a 1 must use neither. A line entering a corner of a 1 exhausts it; entering a 3 forces the two far edges; entering a 2 must exit through the opposite corner, and a sealed corner of a 2 pushes the loop out through its neighbors.
3-3 patternsT5
Two 3s side by side force the shared and outer parallel edges (and stop the shared edge from extending); two 3s on a diagonal force both outer corners.
No early loopT6
An edge that would join the two ends of an existing path may not be drawn if closing the loop there would leave the rest of the puzzle unfinishable.
One-step trialT7
Suppose a single edge is a fence (or an ×) — or a single cell is inside (or outside) — and follow the counting, dot, and parity rules; if that quickly contradicts, it must be the opposite. The workhorse of hard boards.