Contradiction

Expert Used by Smart Hint

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.

Islands that establish the pattern

How it works

When no direct rule settles a connection, suppose an answer and follow it: imagine the highlighted connection carried its smallest (or largest) possible count, and apply the ordinary techniques from there.
If that supposition ends up breaking a rule — an island over or under its number, bridges crossing, or a finished group sealed off — the supposition was wrong.
The connection must then avoid that count. This is the last-resort technique: try the direct ones first.