Hashiwokakero 橋をかけろ

Hashiwokakero (橋をかけろ, Japanese for "build bridges") is a connection puzzle from Nikoli, the Tokyo puzzle publisher that also created Slitherlink and Nurikabe and gave Sudoku its name. The grid holds a scattering of islands, each a circle with a number from 1 to 8. Your job is to draw bridges between them so that every island touches exactly as many bridges as its number. Bridges run straight across or straight down, at most two of them join the same pair of islands, and no bridge may cross another bridge or pass over an island. When the last bridge is in place, the whole archipelago must be one connected network: you could walk from any island to any other.

Outside Japan the puzzle is usually published as Bridges or simply Hashi. The rules fit in a sentence, yet a well set grid still demands real deduction: counting how many bridges an island can still take, spotting pairs that would cut themselves off, and noticing when a drawn bridge blocks the line of sight between two others.

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Start with the islands that are already forced. An 8 must have two bridges in all four directions. A 6 on an edge and a 4 in a corner are the same idea: the island needs two bridges to every neighbour it can see. More generally, whenever an island's number equals twice the number of islands it can see, every one of those links is a double bridge. Draw them first: each bridge you place also blocks the lines it crosses.

Next look for islands with a single visible neighbour: all of their bridges go there. Then apply the "at least one" count. A 3 that can see only two islands must send at least one bridge to each of them, a 5 with three neighbours the same, and a 7 with four neighbours touches every neighbour at least once. These partial bridges are safe to draw before you know where the rest go.

The connectivity rule gives you the isolation deductions. Two 1s never connect to each other, because that pair would be finished and cut off from everything else. A 2 never takes both of its bridges to another 2 for the same reason, and the pattern generalises: a group of islands must never be sealed off while other islands remain outside it. This often decides the last few bridges.

Common mistakes: forgetting that a 2 may be satisfied by a double bridge to one neighbour rather than two single ones; drawing a bridge across a line that another pair still needs; and finishing every number while leaving the grid in two groups. Check highlights islands with too many bridges, finished groups cut off from the rest and bridges that are not in the solution; when nothing is wrong it says how many islands still need work.

Hashiwokakero first appeared in issue 31 of Nikoli’s magazine Puzzle Communication Nikoli, the autumn 1990 issue; its original author had already tried out a prototype of the idea in issue 28. The name is the Japanese imperative “build bridges!”, which is exactly the task. Like Nikoli’s other grid puzzles it later spread abroad under simpler names, most often Bridges or Hashi, and it is now a regular in puzzle books and apps. Nikoli’s own puzzles are made by hand, and its standard Hashiwokakero grids are 10×10 and 18×10.