Bridges strategy: counting, crossing and connecting

Play Bridges

Bridges rewards one question asked over and over: how much room does this island have? Most of the puzzle falls out of that. The rest comes from two rules that are easy to forget — bridges can't cross, and everything has to end up connected.

How the difficulty works

Each tier is set by the hardest kind of reasoning you'll need:

Easy needs only counting, and noticing when a bridge blocks another.
Medium needs at least one step where you rule out a connection because it would cut a group of islands off from the rest.
Hard needs at least one short "what if": assume a bridge, follow it a little way, and watch it fail.

None of them ever needs a long chain of guesses.

Counting: the island's room

Each neighbour an island can reach can take at most two bridges. So an island's room is twice its number of reachable neighbours — less, if a neighbour is a 1, or if a bridge is already in the way. Compare the room with the number:

No slack: the number equals the room. Every connection is a double bridge.
One bridge of slack: the number is one less than the room. Every connection gets at least one bridge.
More slack: nothing yet. Come back once the neighbours have settled.

No slack

The ringed 4 has 2 neighbours, and 2 × 2 = 4. There's no other way to make its number: every connection is a double. From Easy puzzle #1 — play it

The ones to memorise: a 4 in a corner, a 6 on an edge, and an 8 anywhere are always all doubles — provided nothing blocks one of their neighbours.

One bridge of slack

The ringed 3 has room for 4 bridges, one more than it needs. If any connection were left empty, the others couldn't make up 3 on their own, so each gets at least one bridge. From Easy puzzle #8 — play it

Likewise: a 3 in a corner, a 5 on an edge and a 7 in the open always get at least one bridge in every direction. Build those single bridges straight away. They often settle a neighbour, which then settles another.

Counting what's left

As bridges go in, count what each island still needs against the room it still has. A 4 with two bridges already built is really a 2. A neighbour that's complete has no room left to offer. This running count drives most of every puzzle, Easy to Hard.

Crossings

Once a bridge is built, nothing can cross it. Every bridge you add can quietly take a neighbour away from an island on the other side, which shrinks its room — and sometimes leaves it with no slack.

The bold bridge is already certain. The dashed connection between the 3 and the 1 would have to cross it, so it's ruled out — and both islands have less room than they seemed to. From Easy puzzle #2 — play it

Medium: don't cut anyone off

All the islands have to end up joined. So a connection that would complete a small group of islands, leaving them no bridges to spare for the rest of the map, can't be right.

The two ringed 1s could use each other up with a single bridge. But then neither would have any bridges left for the rest of the islands, and the pair would be stranded. So that bridge is ruled out. From Medium puzzle #9 — play it

The classic cases are two 1s facing each other, which can never be joined, and two 2s, which can share one bridge but never two. The same logic works for bigger groups: before closing a loop, check whether the islands inside it still need anything from outside.

Hard: a short "what if"

On Hard puzzles you'll sometimes reach a point where counting and connecting give you nothing. Then pick a connection with only two options — say, one bridge or none — and ask what would happen with one of them. Follow it only a step or two: an island running out of room, two bridges needing to cross, a group being cut off. If it breaks, the other option is the answer.

Suppose there were no bridge between the ringed 2 and 3. Follow the counting from there and, within a few moves, the islands around them can't all be satisfied. So there must be at least one bridge (in red). Try it yourself on the puzzle. From Hard puzzle #1 — play it

The trick is to try the option that looks like it will fail fast. A "what if" that doesn't break within a few moves isn't proving anything; take it back and look elsewhere. Our puzzles never need you to follow one further than that.

Good habits

Corners and edges first. Fewer neighbours means less slack.
Watch the 1s. A 1 can only ever send one bridge, which limits every neighbour it has.
After each bridge, look around it. The islands at both ends, and any connection it crosses, have just changed.
Count at the end. If everything is complete but the game isn't finished, a group is cut off somewhere.