Trial and Error

How it works

1. This is the last rung, and the only one that isn't really a deduction. Only Extreme puzzles are ever built to need it — every easier tier is guaranteed to fall to the techniques above.
2. Pick an undecided edge, and pick it well: one touching a clue or a line you already have, so a wrong choice runs into trouble quickly instead of ten moves later.
3. Commit to it. Say it's a line, then keep solving normally from there — all the usual techniques, as deep as they'll go.
4. This is the whole difference from Contradiction, which only looks a handful of steps ahead. Here you follow the choice as far as it leads, which is why it settles boards Contradiction reports as inconclusive.
5. If it dead-ends — a clue that can't be satisfied, a dot left with exactly one line, or a loop that closes early and strands lines outside it — the guess was wrong. Unwind everything you drew on it and X that edge instead.
6. If it doesn't dead-end but runs all the way to a single closed loop with every clue satisfied, you haven't just tested the guess: that IS the answer, and you're finished.
7. One piece of bookkeeping makes all of this bearable: shade the area you're about to guess in before you commit. The shade is a freeform note, so it costs nothing, and it's the only thing that shows you later how far the guess actually reached — Undo steps back one tap at a time, and by then a wrong guess is a dozen taps deep.