Bridges: counting bridges
Add up the most each neighbor could take. If the number needs all of it, draw every bridge. If it needs all but one, every neighbor gets at least one.
How it works
Each neighbor of an island can take at most two bridges. It takes fewer if its own number is small, if it already has some bridges, or if a bridge in the way blocks it. Add those limits up: that is the most the island could ever get.
Now compare. If the island needs exactly the most, every neighbor gives its full share. If it needs one less, no neighbor can be left out: without that neighbor, the others give too little. So each gets at least one bridge.
In general, a neighbor must give at least the number minus what all the other neighbors could give. That one subtraction is the whole technique. It is elimination with numbers.
One short of the most
When every neighbor could take two bridges, a number one less than the most puts at least one bridge on each side. A 3 with two neighbors, a 5 with three, a 7 with four.
The 5 below has three neighbors, so at most six. It needs five, so each neighbor gets at least one.
Worth memorizing
Most = what every neighbor could take.
Two each, less for small or full neighbors.
Need equals most: draw every bridge.
Need is one less: every neighbor gets at least one.
Practice
0 of 4 rightWhat must be true about the corner 3 on the empty map?
The 5 has three neighbors. How many of them must get at least one bridge?
The 2 in row 5 has one bridge. How many can it still give the 3 on its right?
The 4 has two bridges from the corner 3. Where do its last two go?