Contradiction
Suppose a connection carried its lowest (or highest) possible count and follow the consequences — if an island overflows, bridges cross, or a finished group gets sealed off, that count is ruled out.
How it works
- 1Every direct rule is stuck on this position: no island is short enough, or cornered enough, for the counting rules to bite, and the bridges already drawn rule nothing out.
- 2So suppose an answer and follow it: imagine the connection between the two highlighted islands carried its smallest (or largest) possible count, and apply the ordinary techniques from there.
- 3If that supposition ends up breaking a rule — an island pushed over or under its number, bridges forced to cross, or a finished group sealed off — the supposition was wrong.
- 4The connection must then avoid that count. This is the last-resort technique: try the direct ones first, and only suppose once they all stall.
- Existing bridge
- Islands that establish the pattern