Hashiwokakero Rules: How to Play Hashi, Strategies and Tips

What is Hashiwokakero?

Hashiwokakero, usually shortened to Hashi and often sold as Bridges, is a logic puzzle about building a network. The grid holds a scattering of circled numbers called islands. Your job is to join them with straight horizontal and vertical bridges so that every island has exactly as many bridges as its number, and every island ends up connected to every other.

Unlike Sudoku or Kakuro, you never write a digit. You draw lines, and the puzzle feels more like planning a road map than filling in a grid. Island numbers run from 1 to 8, because an island can have at most four neighbors (up, down, left and right) and at most two bridges to each of them.

The rules take a minute to learn, which makes Hashi one of the friendliest Japanese logic puzzles for beginners. Large grids with dozens of islands and long, tangled sightlines can still hold an experienced solver's attention for a long time.

A short history

The name is Japanese: 橋をかけろ (hashi o kakero) means "build bridges!". The puzzle comes from Nikoli, the Tokyo publisher that also gave Sudoku its name. Hashiwokakero first appeared in Puzzle Communication Nikoli issue 31 in September 1990, and an earlier form of the idea had been printed in issue 28 in December 1989.

English-language publishers have used several names. The Times prints it simply as Hashi, and many books call it Bridges. A mistranslation also gave it the name Chopsticks: hashi written with a different character (箸) means chopsticks, while the 橋 in the title means bridge. In France, Belgium, the Netherlands and Denmark it appears as Ai-Ki-Ai.

Nikoli still publishes the puzzle, and you can see its original presentation on Nikoli's English Hashiwokakero page. For more background, including the solving theory, see the Wikipedia article on Hashiwokakero.

The rules of Hashi

Every Hashi puzzle follows the same five rules:

1. 1Bridges run straightA bridge joins two different islands in a straight horizontal or vertical line. No diagonals and no bends.
2. 2One or two per pairTwo islands can be joined by a single bridge or a double bridge, but never more than two.
3. 3No crossingsBridges may not cross each other, and they may not pass over an island.
4. 4Match the numberThe total number of bridges touching an island must equal the number written in it. A double bridge counts as two.
5. 5Connect everythingWhen you finish, all the islands must form one connected group, so you can travel from any island to any other along the bridges.

The last rule is the one beginners forget. Satisfying every number is not enough if the islands split into two separate groups. A properly made Hashi has exactly one solution, and you never need to guess.

A worked example

Here is a 7×7 puzzle with nine islands, next to its unique solution. Double bridges are drawn as two parallel lines.

Start with the islands that have no choice. The 1 at the top can only see the 4 below it, so it gets a single bridge. The 2 at the bottom can only see the 4 to its right, so it needs a double bridge. That bottom-right 4 has just two neighbors, so both of its bridges must be doubles, and the same is true of the 4 on the left edge. Now the 4 in the middle of the third row already has three bridges (one from above, two from the left) and needs exactly one more, which can only go right, to the 2. Keep counting and the rest follows: the 3 on the left edge needs one more bridge, to its right; the 2 on the right edge sends its last bridge down; and the 5 is completed by a double bridge from the 3 in the middle.

How to play: a step-by-step method

This routine works on almost any Hashi, from a nine-island warm-up to a crowded expert grid:

1. List each island's neighbors. For every island, note which other islands it can see in a straight line up, down, left and right, with nothing in between.
2. Draw the forced bridges first. Look for islands whose number leaves no choice, such as a 1 or a 2 with a single neighbor, or a 4 with exactly two.
3. Add the guaranteed singles. An island whose number is one below its maximum must have at least one bridge to every neighbor. Draw a single to each and decide the doubles later.
4. Tick off finished islands. Once an island's count is met, mark it. Its remaining directions are now closed, which often removes options for the islands around it.
5. Recount after every bridge. Each new bridge blocks any bridge that would cross it. Recheck the neighbors of islands on both sides of it.
6. Check connectivity before you finish. Make sure no group of islands has sealed itself off. If one has, a bridge somewhere is wrong.

Strategies strong solvers use

Almost every Hashi deduction comes from comparing an island's number with how many bridges it could possibly hold. These are the patterns worth learning by heart.

Variations and relatives

Hashi itself changes little from publisher to publisher, but it comes in different flavors and has some close cousins:

• Grid size and density. Beginner puzzles might have ten islands on a 7×7 grid, while expert puzzles pack dozens onto a large rectangle. Density matters as much as size: islands with many long-range neighbors create far more options.
• Unmarked grids. Many published Hashi puzzles do not draw the grid at all, so the islands float on a blank page. The rules are identical, but judging which islands line up takes more care.
• Loop-drawing cousins. Slitherlink and Masyu also ask you to draw lines, but both build a single closed loop instead of a branching network.
• Connectivity cousins. Nurikabe shares Hashi's rule that everything must stay connected, but applies it to a wall of shaded cells rather than a set of bridges.

Common mistakes to avoid

When a Hashi will not come out, the cause is almost always one of these:

• Forgetting connectivity. It is easy to satisfy every number and still end up with two separate groups. Check the whole map, not just the island you are working on.
• Miscounting doubles. A double bridge counts as two toward both of the islands it joins. Recount any island next to a double before you mark it finished.
• Overlooking distant neighbors. A bridge can span a wide stretch of empty water as long as nothing blocks it. Do not look only at the closest islands.
• Drawing a crossing. Two bridges can never cross. When you add a long bridge, immediately cross out any path it cuts.
• Adding a third bridge. Two islands can share at most two bridges. If a count seems to need a third, one of your earlier bridges is in the wrong place.

Why Hashi is worth learning

Hashi is a gentle introduction to thinking in networks. Each deduction mixes a local question (how many bridges does this island still need?) with a global one (will everything stay connected?). That combination is rare among pencil puzzles, and it trains a different habit from the row-and-column scanning of Sudoku or Futoshiki.

The puzzle is also a favorite of mathematicians and programmers. Deciding whether an arbitrary Hashi grid has a solution has been shown to be NP-complete, which is a formal way of saying that no simple shortcut is known for every possible puzzle. Hand-made puzzles avoid that trap by design: every step is reachable by logic, so the pleasure is in spotting the next forced bridge.

If you enjoy drawing lines more than writing numbers, Slitherlink is the natural next puzzle to learn.

Frequently asked questions