The Rules of Hashi

Five rules. Four of them you will never think about again after the first board; the fifth is the puzzle. Here is each of them shown rather than stated.

Rule one: straight, and never over an island

A bridge joins two islands that can see each other along a row or a column, with nothing but empty cells in between. It does not bend, it does not run diagonally, and it does not pass over a third island — if one sits between two others, they are not joined at all, and the pair of shorter connections through it are the only ones available.

This is worth dwelling on for a moment because it is what gives a board its structure. Two islands at opposite ends of a row are only linked if the row between them is empty, so dropping a single island into the middle of a long row cuts one long connection into two short ones and changes what every deduction in that row can say. When you look at a board, what you are looking at is not a set of islands but a set of possible connections between them, and those are fixed before you draw anything.

The consequence for reading a board is practical: an island with three or four visible neighbours has three or four decisions attached to it, and one with a single visible neighbour has exactly one and is therefore already answered. Counting an island’s visible neighbours before anything else is the fastest opening habit there is.

Three islands, two links — and no link between the outer two

The only way to finish it

Rule two and rule three: at most two, and nothing crosses

A pair of islands may be joined by one bridge or by two, drawn as a doubled line, and never by three. That cap is why the largest possible number on an island is eight, and it is why a 3 with only two available directions is decided at a glance: two of its bridges have to go one way and one the other, whichever way round that turns out to be.

No bridge may cross another. On the board that means a horizontal bridge and a vertical one cannot both occupy the same empty cell, so drawing either forbids the other outright. Touching at an island is not crossing — the bridge ends there, and any number of bridges may meet at one island up to its number.

These two rules are the ones you obey without noticing for the first few boards and then start to use deliberately. A drawn bridge is not just an answer, it is a wall: every connection it cuts across is now impossible, the islands at the ends of those connections have less room than they had, and that reduction is frequently what makes the next thing findable.

Illegal — both bridges want the same empty cell

Legal — the vertical link goes unused

Rule four and rule five: exact counts, and one network

Every island must end with exactly as many bridge ends touching it as its number. Not at least, not at most. A 5 has five bridge ends when the board is finished, which given the two-bridge cap means at least three directions are in use.

And the finished board must be connected: one network, with a route between any two islands. This rule says nothing about any particular island, which is precisely what makes it hard to obey by accident. A board can have every number satisfied and be plainly wrong, because it is two separate rings that never meet, and nothing local will tell you that — you have to look at the whole thing.

Almost all of the genuinely hard reasoning in this puzzle comes out of rule five, applied backwards. Instead of checking the finished board, you ask of a bridge you are considering whether drawing it would close a group off from everything else. If it would, it cannot be drawn, and that is a deduction available long before the board is anywhere near finished.

Illegal — every number correct, and two separate networks

Legal — the same four 2s, joined into one

Four things beginners think the rules say

They do not say every pair of visible islands is joined. Most links on a finished board carry nothing at all, and a link that carries nothing is not a mistake, it is the ordinary case. Beginners who try to use every possible connection produce boards where every number is far too large.

They do not forbid a bridge running the entire width of the board. Length is irrelevant; a bridge across eleven empty cells is exactly as legal as one across a single cell, and long bridges are where the crossing rule earns its keep.

They say nothing about diagonals, and there is no diagonal connection in this puzzle at all. Two islands touching corner to corner are not joined and never can be.

And they do not permit guessing. Every board published here was checked to be finishable by argument alone, so a board that will not move is one where something has been missed rather than one that requires a coin to be flipped. That is a promise about the generator, and it is why the generator throws most of what it builds away.

Common questions

Can a hashi bridge cross another bridge?

Never. A horizontal and a vertical bridge cannot share an empty cell, so drawing either one rules the other out completely. Bridges meeting at an island is a different thing and is fine.

What is the largest number an island can have?

Eight. An island sees at most four directions and each direction carries at most two bridges, so eight is the maximum and an 8 is fully decided the moment you see it.

Does every island have to be connected to every other?

Not directly, but there must be a route. The finished network has to be one connected piece, so you can walk from any island to any other along bridges, however long the walk.