How to solve Hashi puzzles: let's build some bridges

Hashi is a logic puzzle made of numbered islands. Your job is to connect them with bridges until every number is satisfied and all the islands belong to one connected network.

The rules are short, but they give you plenty to think about. You can start with a few simple deductions and build from there.

The rules to know

Each number tells you how many bridges must meet that island. Bridges run horizontally or vertically between islands, with no more than two bridges joining the same pair. They cannot cross another bridge or pass through an island.

At the end, every island must connect to the others through the network. Matching all the numbers is necessary, but a group cut off from the rest does not solve the puzzle.

Count what an island can reach

Look along each row and column from an island. Which neighboring islands could it connect to without crossing anything or passing through another island?

An island with only one possible neighbor is a useful place to begin. If its number is 1, it needs one bridge to that neighbor. If its number is 2, it needs two.

Now consider an island numbered 3 with only two possible neighbors. One neighbor alone can supply no more than two bridges, so both neighbors must take part. You may not yet know which connection is doubled, but you know neither can be left out.

That kind of partial deduction is progress. You don't always have to finish an island in one step.

A tiny example with an important lesson

Imagine four islands at the corners of a rectangle. Each is numbered 2, and there are no other islands between them.

Four islands at the corners of a rectangle. Each island is numbered 2. There are no other islands.

A single bridge along each side of the rectangle solves the puzzle. Every island has two bridges, and you can travel through the network to reach all four islands.

Show the solution

One bridge runs along each of the four sides of the rectangle. Every island has two bridges, and all four islands are connected.

What if you put two bridges between the top pair and two bridges between the bottom pair instead? Every island would still have two bridges. But the top pair and bottom pair would be separate groups, with no connection between them.

Two bridges join the top pair of islands and two bridges join the bottom pair. Every island has two bridges, but the top pair and the bottom pair are not connected to each other, so this is not a solution.

That arrangement fails the network rule. It is a useful reminder to keep asking two questions: "Do the numbers work?" and "Can everything still connect?"

Keep checking as the puzzle changes

A new bridge may block a possible connection elsewhere because bridges cannot cross. An island whose number is complete cannot accept another bridge. Each of those changes can make another island easier to solve.

After adding a bridge, take a moment to look around it. Which possibilities have disappeared? Has a nearby island been left with only one way to get the bridges it still needs?

Work with the remaining number of bridges, not just the number printed in the circle. If an island marked 4 already has three bridges, you are now looking for one more.

Take your time with the first few puzzles

It is tempting to draw a connection because it looks tidy. Instead, try to explain why a bridge has to be there. Even a short reason, such as "this island has only one neighbor left," will help you follow your own work.

If something goes wrong, check for an island with too many bridges, a crossing or a group that has become isolated. Those are useful clues about where to reconsider a step.

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