Connect the numbered islands with bridges so every island gets exactly its number, all in one network.
Each island shows a number: the exact count of bridges that must connect to it. Draw straight bridges horizontally or vertically between islands, with at most two bridges between any pair. Bridges can't cross each other, and when you're done every island must be linked into a single connected network.
Hashiwokakero — "build bridges" — appeared in Puzzle Communication Nikoli issue #31 in September 1990, with an earlier form of the same idea published in issue #28 the previous December.
The puzzle is a graph problem in disguise. Numbered islands must be joined by bridges, at most two between any pair, running only horizontally or vertically and never crossing. Each number says exactly how many bridge-ends that island takes.
The constraint that does the real work is the one that sounds like an afterthought: every island must end up connected to every other in a single network. A layout can satisfy all the numbers perfectly and still be wrong for having produced two separate islands of islands.
Bridges is an original build of the island-linking logic puzzle, free to play in your browser with a daily and unlimited practice. Made in-house at PlayPendium.