The history of Bridges

Bridges is one of the quieter success stories of the Japanese puzzle world: no craze, no headlines, just a puzzle that people who find it tend to keep playing for years.

From a Japanese puzzle magazine

The puzzle comes from Nikoli, the Tokyo publisher that also gave Sudoku its name. Its magazine has long run on puzzles sent in by readers: someone invents a new kind of grid, the editors try it, and if it's fun to solve it runs again with puzzles by other readers, until the best ideas become classics. Bridges first appeared there around 1990, under the Japanese name hashi o kakero, "build bridges!", usually written Hashiwokakero. In English it's often called Hashi for short.

Nikoli's puzzles share an outlook. They have simple rules that fit in a few lines, a single solution, and a solving path that should feel like a series of small insights rather than trial and error. Bridges fits that perfectly: you can explain it in thirty seconds, and still be surprised by it a hundred puzzles later.

A puzzle about bridges, 250 years earlier

There's a pleasing echo in the name. In 1736, Leonhard Euler was asked about the city of Königsberg, whose seven bridges joined two river banks and two islands. Could you walk through the city crossing each bridge exactly once? Euler proved you couldn't, and in doing so he started a new branch of mathematics, graph theory: the study of points joined by lines.

A Bridges puzzle is a graph-theory problem too. The islands are the points, the bridges are the lines, and the numbers fix how many lines meet at each point. Add the rule that lines can't cross and that the whole thing has to hang together, and you have the puzzle. That's also why one small fact always holds: add up all the numbers on the board and you'll get an even total, because every bridge has two ends.

Harder than it looks

For a computer, solving Bridges puzzles in general is difficult. In 2009 the computer scientist Daniel Andersson showed that the problem is NP-complete: as the board grows, no known method can solve every puzzle quickly, and checking all the possibilities becomes hopeless.

That's no obstacle for a 7×7 or even an 11×13 board, which a program can work through in an instant. The craft is in making puzzles that a person can solve with pleasure.

Variations

Some computer versions let you raise the limit to three or four bridges between a pair of islands. That makes the counting looser, so the connection rule has to do more of the work. The original, with at most two bridges and one connected network, is still the favourite, and it's the one you'll find here.