A Bridge Is Also a Wall

The rule against crossings is trivial to obey and easy to forget to use. Turning it from a constraint into a tool is exactly what medium means here.

Reading a drawn bridge backwards

Every bridge on the board occupies the empty cells it runs over, and no other bridge may use them. So the moment you draw one, every link that would have crossed it is impossible — not unlikely, not unattractive, impossible — and can be struck out immediately.

The reason to do it as a deliberate pass rather than as it occurs to you is that the payoff is indirect. Striking a link out places nothing and changes nothing visible. What it changes is the arithmetic at both ends: two islands that had four directions now have three, their maximum capacity has dropped, and one of them may now be a full house. The crossing rule almost never finishes anything by itself and almost always sets up the move that does.

This is why long bridges matter out of proportion to their length. A bridge over one empty cell can block at most one link. A bridge across seven cells can block seven, and on a large board a single long double bridge sometimes eliminates a quarter of the remaining uncertainty in one stroke.

Before — a bridge already runs the width of row 4

After — the link it cuts across is dead

The two 3s in row 4 were joined a move ago, and that bridge runs across column 6. The 2 at row 2, column 6 and the 2 at row 6, column 6 can see each other down that column, but any bridge between them would have to cross it — so the link is struck out and both of those 2s now have one fewer direction to work with.

what the reasoning read what it decides what it wrote

Where to look, and what people miss

After drawing any bridge, follow it end to end and check every crossing line. That is the whole discipline, and it takes seconds. The failure mode is not ignorance of the rule — everybody knows bridges do not cross — it is that a bridge drawn twenty moves ago stops being noticed. It is furniture by then, and the link it forbids looks as open as any other.

Vertical bridges are forgotten more than horizontal ones, in the same way that columns are forgotten more than rows in every grid puzzle ever devised. A board that will not move where every horizontal bridge has been checked is very often one vertical crossing away.

It is also worth checking crossings the other way round when you are considering a link rather than drawing one. If a link you are thinking about would cross something already built, it is not a candidate at all and does not belong in whatever arithmetic you were about to do. Filtering the impossible out before you count is most of what makes the next technique bearable.

Why this is exactly one level up

A medium board here is one the solver finishes with island counting and the crossing rule and nothing harder. It is a real threshold rather than a gradual one: an easy board never needs a single crossing to be struck out to keep moving, and a medium board contains at least one moment where nothing at all will happen until one is.

The moment is usually easy to recognise once you know to look for it. Every island either has room to spare or is finished and closed out, nothing is forced, and there is a long bridge sitting across the middle of the board that you drew ten moves ago and have stopped seeing.

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

Do bridges that meet at an island count as crossing?

No. A bridge ends at an island, so any number of them may meet there up to the island’s number. Crossing means two bridges wanting the same empty cell between islands.

Why does striking out a link help if it places nothing?

Because it reduces the capacity of the island at each end. An island with less room is often a full house that was not one before, so the crossing rule is what sets up the cheaper techniques rather than what finishes anything.

Which crossings do solvers most often miss?

The ones made by a bridge drawn long ago, especially a vertical one. A bridge stops being noticed once it is part of the picture, and the link it forbids goes on looking open.