Hashiwokakero (Bridges)

Hashiwokakero, commonly known as Bridges or Hashi, is a logic puzzle played on a grid containing numbered circles called "islands." The player must draw horizontal and vertical lines (bridges) between islands so that each island has exactly as many bridges connected to it as its number indicates. Bridges cannot cross each other, at most two bridges can connect any pair of islands, and all islands must be connected into a single network. The puzzle requires careful planning to avoid dead ends and ensure global connectivity.

Rules

1. The grid contains islands — circles with numbers from 1 to 8.
2. Connect islands with horizontal or vertical bridges (straight lines).
3. Bridges cannot cross other bridges or pass through other islands.
4. At most two bridges may connect any pair of adjacent islands (in the same row or column with no island between them).
5. The number on each island indicates exactly how many bridges must connect to it.
6. All islands must be connected — you must be able to travel from any island to any other island via bridges.

Strategies

• Maximum Bridges: An island with value 8 must use two bridges to each of its four neighbors. An island with value 4 in a corner (two neighbors) must use two bridges to each neighbor. Generally, if an island's number equals twice the number of its available neighbors, all connections must be doubled.
• Near-Maximum: An island with value 7 has four neighbors and must connect to all of them — at least one bridge to each, with three of them doubled. Similarly, a value-3 island with only two neighbors must connect to both (at least one bridge each, one doubled).
• Isolation Prevention: If connecting or not connecting a bridge would isolate a group of islands from the rest, that choice is forced. Always check whether a potential bridge would cut off part of the network.
• Single Neighbor: If an island has only one possible neighbor left to reach its required count, all remaining bridges must go to that neighbor.
• Parity: If an island needs an odd number of remaining bridges and has only two available neighbors, at least one bridge must go to each (you cannot distribute an odd number evenly between two options using only 0 or 2).
• Near-Complete Islands: When an island has all but one of its bridges placed, the last bridge's destination is often forced.

Hashiwokakero was created by the Japanese puzzle publisher Nikoli and first appeared in their puzzle magazines in 1990. The name is Japanese for "build bridges" (橋をかけろ), reflecting the puzzle's core mechanic. Nikoli shortened the name to "Hashi" for convenience, and the English name "Bridges" became the most common international designation.

The puzzle emerged during Nikoli's most prolific period of puzzle invention, alongside other classics like Slitherlink and Masyu. Nikoli's design philosophy — simple rules, no arithmetic, pure logic — is perfectly embodied in Hashiwokakero. The rules are intuitive and visual, yet the puzzles can be fiendishly difficult.

Hashiwokakero gained international recognition through the World Puzzle Championship, where it has been a regular fixture since the 1990s. The puzzle was also popularized by Simon Tatham's Portable Puzzle Collection, an open-source set of single-player puzzle games available on multiple platforms, which introduced Bridges to a wide audience of computer users.

The puzzle's appeal lies in its spatial reasoning requirements. Unlike grid-filling puzzles where each cell is independent, Hashiwokakero demands that the solver think about the global structure — ensuring connectivity while respecting local number constraints. This combination of local and global reasoning gives the puzzle a distinctive character. Today it appears in puzzle books, newspapers, and apps worldwide, and remains one of Nikoli's most popular original creations.