Islands With No Choice

Both moves look at one island and nothing else. Between them they finish every easy board on this site, and they open every board at every level.

The full house: no room to do anything else

An island has a number and a set of directions it can see. Multiply the directions by two — the most any one of them can carry — and if the answer equals the number, every one of those links is doubled and there is nothing to decide. A 4 seeing two neighbours, a 6 seeing three, an 8 anywhere on the board: all completely determined before you have looked at anything else.

The same argument covers the narrower case that a beginner meets first. An island with exactly one link left open has no choice about it either, whatever its number: everything it still owes has to go down that one link. That is why the two are one technique rather than two — they are the same question, which is whether this island has any freedom at all, and the answer comes from the island alone.

It really is the island alone. No memory of the rest of the board, no arithmetic across anything, no consideration of what else might want that link. Two numbers compared, and up to four bridges placed. It is the cheapest move in the puzzle by a very wide margin, and on a 15×15 easy board it accounts for most of the placements outright.

Before — the 4 in the top row sees only two islands

After — both of its links are doubled

The 4 at row 1, column 2 can see the 4 to its right and the 2 below it, and nothing else — the board offers it two directions. Two directions carry at most four bridge ends between them, which is exactly what it needs, so both links are doubled and four bridges go down from a single glance at one island.

what the reasoning read what it decides what it wrote

The spent island: closing what is left

The mirror image, and the move that keeps a board moving. When an island already has every bridge its number asks for, it cannot accept another, so every link still open around it can be ruled out entirely. Nothing new appears on the board when you do this, which is precisely why people skip it.

Skipping it is the commonest reason a bridges board feels stuck for no reason. Closing a link takes capacity away from the island at the other end, and an island with less capacity is very frequently a full house that was not one a moment ago. The cascades that make this puzzle satisfying are almost entirely made of this: an island fills, its spare links close, the island beyond becomes forced, it fills, and so on across the board.

On paper, mark the closed links. A ruled-out link that is not written down will be reconsidered four or five times before the board is finished, and each reconsideration is a chance to make a mistake. The boards on this site draw a closed link as a faint dashed line, which is exactly the notation to copy if you are working with a pencil.

Before — the 2 at the foot of that bridge is already full

After — its remaining link is struck out

The double bridge from the previous move gave the 2 at row 6, column 2 both of the bridge ends it asked for. It cannot take another, so the link running right from it to the next 2 is ruled out and drawn as a dashed line. Nothing was built here, and that is the point: the island at the far end now has less room than it had.

Why these two are the whole of easy

An easy board on this site is one the solver finishes using nothing but these two moves. That is the definition rather than an approximation of it: the solver takes the cheapest available step every time, and if it never needed anything above the second rung, the board is easy.

Which also means an easy board is guaranteed to be finishable by a reader who knows only what is on this page. If yours has stalled, either an island that is already finished still has open links attached, or a full house has gone unnoticed somewhere — and the second is much rarer than the first.

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

What is a full house in a bridges puzzle?

An island whose number equals twice the directions it can see, so every one of its links has to be doubled. A 4 with two neighbours, a 6 with three and any 8 are the cases you will meet.

Why should I mark links I have ruled out?

Because a ruled-out link is a result, and an unrecorded result gets reconsidered repeatedly. Marking them is also what makes the next full house visible, since it is the reduction in an island’s room that creates one.

Can these two moves finish a whole board?

On an easy board here, yes — that is what easy is defined to mean on this site. From medium upwards they will stall somewhere and the crossing rule takes over.