How to solve Bridges: connecting every island

The rules

• Bridges run straight, horizontally or vertically, between two islands.
• Each island's number is how many bridges touch it.
• Two islands share at most two bridges.
• Bridges never cross each other or pass over islands.
• In the end, all islands are connected into one group.

Count the neighbours

Each island can connect only to the nearest island in each of its four directions, so it has at most four neighbours and can take at most two bridges from each. Compare its number with what's possible:

• An island with as many bridges as its neighbours can give takes the maximum from all of them: an 8 gets double bridges in all four directions, a 6 on an edge (three neighbours) and a 4 in a corner (two neighbours) do too.
• An island just one short of that maximum (a 7 with four neighbours, a 5 with three, a 3 with two) gets at least one bridge to every neighbour.
• An island with one neighbour takes all its bridges from it.

Don't close a group early

All islands must end up connected, so a set of bridges that seals off a small group of islands is wrong, even if every number in it is satisfied. The classic case: two 1s next to each other can't be joined, because they'd form an isolated pair. Two 2s can't take a double bridge between them for the same reason.

Use this to decide: if one choice would complete a closed group, take the other.

Use crossings

Bridges can't cross. When a bridge is certain, it blocks every potential bridge that would cut across it. That can leave an island with fewer neighbours than before, which brings you back to counting.

A solving routine

1. Draw the bridges for islands whose number equals their maximum.
2. Give one bridge to every neighbour of islands one short of their maximum.
3. Finish islands with a single remaining neighbour.
4. Remove options that would cross a fixed bridge or seal off a group.
5. Recount and repeat until every island is satisfied and connected.

Practise

On Puzzimo, tap an island and then a neighbour (or drag between them) to add a bridge; tap again for a double bridge and a third time to remove it. Undo is free. Boards come in 7×7, 10×10 and 13×13, with a 7×7 daily challenge, and every puzzle has exactly one solution.

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