Bridges: no island cut off
Every island ends up in one network. So a bridge that would leave a group of full islands joined only to each other can’t be there.
How it works
The last rule says every island joins one connected network. That rule does work long before the end.
Suppose a bridge makes a group of islands that are all full. Full islands take no more bridges, so the group can never reach anything else. If other islands are still waiting, that is a contradiction, and the bridge is out. A group like that is an isolated group.
The smallest cases are worth memorizing. Two 1s can’t share a bridge. Two 2s can’t share a double bridge. And the rule turns around: if a group has only one way out left, that bridge must be there.
The shape of the argument
Suppose the bridge is there. Then the group is full, so it has no more bridges. But every island must join one network, and some islands are outside the group. Both can’t be true, so the bridge is not there.
One way out
Look at any group joined by bridges you are sure of. If only one of its sides can still take a bridge, and islands wait outside, that bridge must be there. Cut it and the group is isolated.
Worth memorizing
Two 1s never join.
Unless they are the only two islands.
Two 2s never share a double bridge.
A group with one way out must use it.
Practice
0 of 4 rightCan the two 1s share a bridge?
What is the most bridges the two 2s can share?
Which of these bridges would cut off a group?
Five islands are joined. Which bridge is their one way out?