A Group That Must Not Close

The connectedness rule looks like something you check at the end. Used backwards it is the sharpest deduction in the puzzle, and it is what expert means here.

Closing off: the argument that rules a bridge out

Ask what a link would do if it carried a particular number of bridges. Work out which islands would then be joined together into one group, and ask whether that group would have any capacity left over to reach anywhere else. If it would not — every island in it satisfied, no room anywhere for a bridge to the outside — the group is sealed. And a sealed group that is not the whole board can never be part of a correct answer, so the link cannot carry that number.

The famous case is two 1s that can see each other. Joining them satisfies both at once, and the pair is then a closed island of two with no capacity for anything else. Unless those two are the entire board, the link between them is never used, and you can strike it out on sight. The same reasoning kills the double bridge between two 2s, and a great deal else besides once boards get large.

What makes this hard is that the argument is about a group rather than an island, and the group has to be worked out before the question can even be asked. That is why it is the top rung: everything below looks at one island and its links, and this one asks you to trace a component.

Before — two 1s can see each other along the top row

After — the link between them is struck out

The 1 at row 1, column 4 and the 1 at row 1, column 6 have nothing between them, so a bridge could run across. But one bridge would satisfy both of them at once, and the pair would then be a closed network of two islands with no capacity left to reach the other six. That cannot be part of any correct answer, so the link is ruled out — and each 1 is now forced downwards.

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One way out: the argument that forces a bridge

The mirror image, and nobody learns one of these without the other, which is why they share a rung and a page. Take a group of islands that are already joined to one another and look at every link leaving it that could still carry something. If there is exactly one, that link has to carry at least one bridge — the group must reach the rest of the board somehow, and this is the only door left open.

It is rarer than the closing-off form in practice, largely because the cheaper techniques usually settle a group’s last remaining link before anyone gets round to counting the doors. But it is the form that most often breaks a board that has been stuck for ten minutes, precisely because it looks at something no other technique looks at.

Both arguments become available only once a board is part-built, since neither has anything to say while every island is still its own group. That is what gives an expert board its characteristic shape: it moves along under ordinary arithmetic like anything else, arrives two-thirds finished with nothing forced anywhere, and then turns on a single observation about what would be cut off.

Why this is the top of the published ladder

Above it there is only the assumption — draw something, follow it until the board breaks, rub it out — and that will finish any board at all, which is exactly why it cannot be a difficulty level. A scale whose top rung fits everything measures nothing.

So expert here means precisely this technique and no more: somewhere in the board there is a link that nothing else will decide, and the reason it is decided is that using it would strand a group. Boards that need anything beyond that are discarded during generation rather than sold as a harder level.

On a small board expert is genuinely hard to produce. Seven columns often do not give a group enough room to seal itself off in the first place, which is why a 7×7 asked for at expert quite frequently comes back labelled hard — that is the label the solver measured, and it ships under that rather than under the one requested.

Where this sits on the ladder

Rung 1Full house
An island with one open link, or with only just enough room, has no choice at all.
Rung 2Spent island
An island that already has every bridge it asked for closes all its other links.
Rung 3Blocked crossing
A link that would cut across a bridge already built can never be used.
Rung 4Spare capacity
Take one link away and ask what the rest could carry; the shortfall is forced down it.
Rung 5Isolation
A group cannot seal itself off, and a group with one way out has to use it.
Rung 6Assumption
Draw a bridge, follow it until something breaks, and rub it out again.

Common questions

Can two 1s next to each other ever be joined in hashi?

Only if they are the entire board. On any larger board joining them satisfies both at once and seals them off from everything else, which breaks the rule that the finished network must be connected.

What is the one-way-out argument?

If a group of islands already joined together has exactly one remaining link that could reach outside it, that link must carry at least one bridge, because the group has to connect to the rest of the board somehow.

Why does connectivity only start mattering late in a board?

Because it reasons about groups, and early on every island is its own group with plenty of room. It has nothing to say until several bridges have joined islands into components worth reasoning about.