Hashi puzzles, and how Lacey’s Web borrows from them

What a Hashi puzzle is

A Hashi puzzle is a grid with numbered circles on it, called islands. You draw bridges between them. The rules:

• Bridges run straight, across or down, between two islands.
• Two islands can have one bridge between them, or two, never more.
• Bridges can’t cross each other or pass over an island.
• Each island’s number is how many bridges touch it.
• When you’re done, every island is joined to every other, as one network.

That last rule is the one that makes Hashi more than counting. You can satisfy every number and still lose, because the islands have broken into two groups that never meet.

Where it came from

Hashi’s full name is Hashiwokakero, from the Japanese for “build bridges!”. It comes from Nikoli, the Japanese puzzle publisher, and first ran in its magazine, Puzzle Communication Nikoli, around 1990.

In English it goes by several names. Simon Tatham’s free puzzle collection calls it Bridges. Conceptis’s app calls it Hashi: Bridges. You’ll also see Chopsticks, because in Japanese “hashi” can also mean chopsticks.

Large Hashi puzzles are hard in a precise sense. In 2009 the computer scientist Daniel Andersson proved that solving Hashiwokakero in general is NP-complete, which roughly means no one knows a shortcut that solves every big one quickly. That’s why good Hashi puzzles are made and checked with care: the point is a puzzle you can reason through, step by step, with one answer at the end.

How to solve a Hashi puzzle

Almost every step comes from one question: how many bridges can this island still take, and where can they go?

1. Count the neighbors. An island can reach at most four others (one each way), with at most two bridges each. So it can hold at most twice as many bridges as it has neighbors.
2. Full house. If the number equals that maximum, every bridge is a double. A 4 in a corner (two neighbors) gets two bridges each way. A 6 with three neighbors, or an 8 with four, the same.
3. One short of full. If the number is one less than the maximum, every neighbor gets at least one bridge. A 3 with two neighbors needs one bridge to each, for a start.
4. Don’t wall in a pair. Two 1s can’t join each other: they’d be full, and cut off from everything else. The same goes for two 2s with a double bridge between them. Rule these out early.
5. Cross off what’s finished. Once an island has its number, it takes no more. That often leaves its neighbors with only one place to go.
6. Watch the crossings. Every bridge you draw blocks any bridge that would cross it. A long bridge can settle three other islands at once.

When nothing obvious is left, look at the whole map. Which groups are still apart, and which bridges are the only ones that could join them? The connection rule settles the last few bridges more often than the numbers do.

How Lacey’s Web borrows from Hashi

Lacey’s Web is our daily puzzle, and it grew from the same idea. Grandma swept Lacey the spider’s web, and only the knots are left. Each knot shows how many threads it held. You spin the threads back so that every knot has its number and all the threads make one web.

What’s the same:

• Each number is how many connections touch that knot.
• Everything has to end up as one connected web. A set of threads that closes itself off is wrong, even if every number in it is right.
• It’s solved by logic, with one answer. Every daily web is checked that way before anyone plays it: a solver mends it from the numbers alone, using a player’s own rules, and a web it can’t finish is thrown out. (How we make puzzles with exactly one answer has the details.)

What’s different:

• The lines are already drawn. In Hashi you find the bridges in open space. In Lacey’s Web every possible thread is a faint line on the web: the spokes running out from the hub and the rings running round. You decide which of them are threads.
• One thread per line. There are no double bridges. Each line is a thread or it isn’t.
• No crossings to worry about. The web’s lines only meet at knots, so a thread never blocks another.
• The frame threads are given. Each outer knot already has a thread out to the attic frame, and it counts toward that knot’s number.
• “?” knots. On the daily web, a few knots show “?” instead of their number. A “?” knot has no number to meet. It only has to join the web.

Hashi habits that work in Lacey’s Web

Most of the Hashi tips carry straight over, a little simpler, since a line holds one thread at most:

• Full house becomes “spin them all.” If a knot’s faint lines are exactly what it still needs, spin every one.
• Cross off what’s finished becomes “✕ the rest.” When a knot has its number, mark its other lines ✕.
• Don’t wall in a pair becomes “keep it one web.” Don’t spin a thread that would close off a group whose knots are all full, and don’t ✕ a line that’s the only way to reach part of the web.
• Look at the whole map. On the harder days, try one line in your head: if spinning it would force a knot past its number two or three moves later, it’s a ✕.

For a full walk-through of a practice web, see our Lacey’s Web guide.

Where to play Hashi

• Simon Tatham’s Portable Puzzle Collection has Bridges, free, in a web browser or as a download. You can set the size and difficulty, and play as many as you like.
• Conceptis makes Hashi: Bridges, an app for iPhone and iPad.
• Puzzle by Nikoli W Hashiwokakero is a paid game in the Microsoft Store, made by HAMSTER Corporation under the Nikoli name.
• Lacey’s Web is one new web a day, free in your browser, about two minutes on a weekday and a bigger web on Sunday. Same puzzle for everyone, with a streak and a time to beat.

Questions people ask