The History of Hashi (Hashiwokakero)

Niels Glodny · 3 November 2026

Hashi comes from Nikoli, the Tokyo puzzle publisher that also gave Sudoku its name. It first appeared in 1990 and has since been printed under several names in Europe. In 2009 a computer scientist proved that it is hard in a precise mathematical sense.

Nikoli, 1989 and 1990

Nikoli was founded in 1980 by Maki Kaji and named after a racehorse that won the Irish 2,000 Guineas that year. Its quarterly magazine, Puzzle Communication Nikoli, is where many of the Japanese logic puzzles known today were first printed.

Hashiwokakero appeared in issue 31 of the magazine, in September 1990. An earlier form of the puzzle had been printed in issue 28, in December 1989.

The name

橋をかけろ (hashi o kakero) is Japanese for “build bridges!”, phrased as an order. Outside Japan the name is usually shortened to Hashi, or translated as Bridges.

It has also been published in English as Chopsticks, from a mistranslation: spoken Japanese hashi can mean bridge (橋) or chopsticks (箸), depending on the character. In France, Denmark, the Netherlands and Belgium the puzzle is published as Ai-Ki-Ai, and The Times has printed it as Hashi.

Computers and Hashi

Simon Tatham’s Portable Puzzle Collection, a free set of puzzle games for many platforms, includes it under the name Bridges.

In 2009, Daniel Andersson published a two-page proof that Hashi is NP-complete. In practical terms, no known method solves every Hashi puzzle quickly, and the work needed can grow very fast with the size of the board. Sudoku, generalised to larger grids, is in the same class. The proof is about the worst cases. Boards of the size people play are solved by a computer in a fraction of a second.

In 2019, Leandro Coelho, Gilbert Laporte, Arinei Lindbeck and Thibaut Vidal published a set of 1,440 Hashi puzzles for testing solvers, along with a solver based on integer programming. It adds the rule that all islands must connect only when a candidate solution breaks it, and it reliably solved hard puzzles with up to 400 islands.

This game uses a related idea with a different kind of solver, a SAT solver, both to solve puzzles and to check that every generated puzzle has exactly one solution. How the solver works explains it.

Sources

• Hashiwokakero, Wikipedia, for the first issues and the names abroad.
• Nikoli (publisher), Wikipedia.
• Hashiwokakero, Nikoli’s rules page.
• Bridges, Simon Tatham’s Portable Puzzle Collection.
• Daniel Andersson, Hashiwokakero is NP-complete, Information Processing Letters 109(19), 2009.
• Coelho, Laporte, Lindbeck, Vidal, Benchmark instances and branch-and-cut algorithm for the Hashiwokakero puzzle, 2019.