Hashiwokakero: The Story of the Bridge Puzzle

Hashiwokakero, the bridge puzzle also called Hashi or Bridges, was created in Japan for the puzzle magazine Puzzle Communication Nikoli. It first appeared in issue 31, in September 1990, after an earlier version in issue 28 in December 1989. The inventor was a reader who wrote under the pen name Lenin and, according to Atlas Obscura, took the idea from chemistry's diagrams of bonds between atoms. The name means "build bridges!".

Nikoli, the magazine where readers invent puzzles

Hashiwokakero comes from Nikoli, the Japanese puzzle publisher that gave sudoku its name. Nikoli says on its company page that it published a puzzle magazine called NIKOLI in August 1980 and set up the company at the same time, and that it was the first puzzle magazine in Japan.

What makes Nikoli unusual is where its puzzle types come from. Many of them are invented and refined by readers, who send in new kinds of puzzle and work on each other's ideas in the magazine's pages. Atlas Obscura, which ran Nikoli puzzles in its Puzzle Monday series, describes the magazine as a place where "many of puzzle designs are conceived and improved by dedicated readers". Hashiwokakero is one of those reader-made puzzles, and one of the older ones still running.

Sudoku itself has a different story: it began as Number Place, published in the United States by Dell Magazines in 1979, and Nikoli introduced it in Japan in 1984. That history, including how it reached The Times, is in where sudoku came from.

Lenin, the reader behind the puzzle

Hashiwokakero was created by a Nikoli reader and puzzle constructor who used the pen name "Lenin" (れーにん, written renin in romanised Japanese). Atlas Obscura notes that the name has nothing to do with the Russian revolutionary: the constructor's real name, written in Chinese characters, can be read as "Lenin".

The same constructor is credited with Nurikabe, the puzzle of walls and islands drawn on a grid. Wikipedia's article on Nurikabe) names "renin" as its developer and dates its first appearance to issue 33 of Nikoli, in March 1991, a few months after Hashiwokakero. Atlas Obscura also credits Lenin with Slitherlink, though Wikipedia's account of Slitherlink's origin says Nikoli's editor combined two reader contributions to create it, so the full share of credit there is less clear.

We found little else about the constructor in public sources. What is clear is that one reader's ideas account for two or three of the best-known puzzles Nikoli has published.

From chemical bonds to bridges

Hashiwokakero began as a puzzle about atoms, not islands. According to Atlas Obscura, Lenin was inspired by the diagrams that show molecular bonding in chemistry, where atoms are joined by single and double lines. An earlier version he submitted was called Ketsugo-shu, which Atlas Obscura translates as "Atomic Bonding". When he sent in the final version, he changed the picture in the title from bonds to bridges.

The dates of those two versions are given by Wikipedia and repeated by the puzzle publisher Conceptis: an earlier form in issue 28 (December 1989) and the puzzle itself in issue 31 (September 1990).

The chemistry is easy to see once you know to look for it. An island's number works like an atom's valence: the number of bonds it must make. Here are two small molecules drawn as Hashi puzzles that solve exactly like the real structural formulas.

Methane drawn as a Hashi puzzle: a central 4, like carbon, with a single bridge to each of four 1s, like hydrogen. - and | are single bridges. This is an illustration, not a historical puzzle.

Carbon dioxide drawn as a Hashi puzzle: a 4 in the middle with a double bridge to a 2 on each side, like the double bonds in O=C=O. = is a double bridge.

The analogy has limits. Chemistry has triple bonds, while Hashi stops at two bridges between any pair of islands. And Hashi's rule that every island must join one network has no direct counterpart in a drawing of a single molecule, where that is true anyway.

What "Hashiwokakero" means

Hashiwokakero is a short Japanese sentence: 橋をかけろ, hashi o kakero, "build bridges!" It is a command, which suits a puzzle that tells you what to do.

Part	Japanese	Meaning
hashi	橋	bridge, or bridges
o	を	marks the object of the verb
kakero	かけろ	command form of kakeru, "to build (a bridge)"

The verb kakeru, written 架ける when it means building a bridge, is defined in the Jisho dictionary as "to suspend between two points; to build (a bridge, etc.)". The o is written with the character を but pronounced o, which is why you see both "Hashiwokakero" and "Hashi o kakero" in English.

The word hashi has a well-known double meaning. Written 箸 instead of 橋, hashi means chopsticks, and Wikipedia notes that the puzzle has appeared in English as "Chopsticks" because of that mistranslation. The two words are written with different characters, but in romanised spelling they look the same.

One puzzle, many names

Hashiwokakero is published under several names, depending on the country and the publisher. These are the ones recorded by Conceptis and Wikipedia.

Name	Where it is used
Hashiwokakero	the original Nikoli name, in Japan
Hashi	The Times in the UK, and many apps and books
Bridges	the common English name
Chopsticks	an English name based on the mistranslation of hashi
Ai-Ki-Ai	France, Denmark, the Netherlands and Belgium

Puzzle Picnic calls the game Bridges, with Hashi in brackets. The rules are the same under every name.

How Hashi reached newspapers and screens

Outside Japan, Hashi is best known from newspapers and puzzle books, the route sudoku took. When The Times in London began printing sudoku on 12 November 2004, it set off a newspaper craze that spread rapidly to other papers. The Times has also published the bridge puzzle as Hashi, and in 2011 Times Books brought out a collection, The Times Hashi, compiled by Puzzler Media. The book's description calls Hashi, after sudoku, "one of the most popular Japanese logic puzzles".

The puzzle also suited computers well. One of the best-known free versions is Bridges in Simon Tatham's Portable Puzzle Collection, a set of small logic games for many platforms. Its documentation credits the puzzle to Nikoli and says Bridges was contributed to the collection by James Harvey.

In 2022 Atlas Obscura featured Hashiwokakero in its weekly puzzle series, noting that at 32 years old it was one of the older puzzles still in Nikoli's magazine. Today it is common in puzzle apps, including Puzzle Picnic, where it is one of six games and comes in four sizes from 7 x 7 to 11 x 14.

What mathematicians found

Hashiwokakero is simple to state but hard in general, and computer scientists have proved it. In 2009 Daniel Andersson published a short paper in Information Processing Letters, "Hashiwokakero is NP-complete". It shows that deciding whether a Hashi board has a solution belongs to the class of NP-complete problems, for which no method is known that stays fast as boards grow very large. The proof works by turning a known hard problem, about routes through a special kind of graph, into Hashi puzzles.

That result is about huge, arbitrary boards. The puzzles people actually solve are small and carefully made, and they can be solved by chains of logical steps. Researchers have studied those steps too:

In 2012 Reza Firsandaya Malik and colleagues described a program that solves Hashi with named human-style techniques (Just Enough Neighbor, One Unsolved Neighbor, Few Neighbor, Leftovers and Isolation) combined with a depth-first search for what those techniques leave open.
In 2019 Leandro Coelho, Gilbert Laporte, Arinei Lindbeck and Thibaut Vidal published benchmark puzzles and a branch-and-cut algorithm for Hashi. They tested it on 1,440 puzzles and solved instances with up to 400 islands.

For a solver at home, the useful lesson is the first one: most of a Hashi puzzle falls to a handful of repeatable deductions. They are set out in order in Hashi strategies.

The rules then and now

Nikoli's rules for Hashiwokakero are short. The publisher's English rules page gives four: connect the islands with as many bridges as the number on each, no more than two bridges between two islands, bridges cannot cross islands or other bridges, and the bridges form a continuous link between all the islands. Bridges run only horizontally or vertically.

Those four rules explain the puzzle's familiar numbers. An island can face at most four neighbours, and each can take at most two bridges, so no island ever shows more than 8. The connectivity rule is what makes the puzzle more than arithmetic: it rules out moves that would satisfy the numbers but leave a group of islands cut off, and it is the source of some of the most satisfying deductions in the game, covered in the connectivity rule explained.

Variants exist mostly on computers, where changing a rule costs nothing. Simon Tatham's version lets you raise or lower the two-bridge limit and add a rule that forbids loops. Puzzle Picnic keeps Nikoli's original rules.

Year	Event
1980	Nikoli publishes its first puzzle magazine, in August
1989	An early form of the puzzle appears in Nikoli issue 28, in December
1990	Hashiwokakero appears in Nikoli issue 31, in September
1991	Nurikabe, by the same constructor, appears in issue 33, in March
2004	The Times starts printing sudoku, on 12 November
2009	Daniel Andersson proves Hashiwokakero is NP-complete
2011	Times Books publishes The Times Hashi
2019	A benchmark of 1,440 Hashi puzzles is published by Coelho and colleagues

Frequently asked questions

Who invented Hashiwokakero?

A reader of the Japanese magazine Puzzle Communication Nikoli who used the pen name Lenin. The same constructor is credited with the puzzle Nurikabe. Nikoli published Hashiwokakero in issue 31, in September 1990.

What does Hashiwokakero mean in English?

It means "build bridges!", as a command. Hashi is bridge, o marks the object, and kakero is the command form of the verb for building a bridge.

Why is Hashi sometimes called Chopsticks?

Because hashi written with a different character, 箸, means chopsticks. In romanised spelling the two words look the same, and the puzzle has been published in English under that mistaken name.

Is Hashi older than sudoku?

No. Number Place, the puzzle that became sudoku, was first published in the United States in 1979, and Nikoli introduced it in Japan in 1984. Hashiwokakero first appeared in 1990. See where sudoku came from for that story.

Are Hashi, Bridges and Ai-Ki-Ai the same puzzle?

Yes. They are names for the same puzzle in different places, with the same rules: numbered islands, straight bridges, at most two per pair, no crossings, and every island joined in one network.

Where to go next

If the history has made you want to try one, how to play Bridges covers the rules and walks through a small first puzzle. When you want to get quicker, Hashi strategies sets out the deductions in order, and every Bridges article is on the Bridges topic page.