Bridges looks like a counting puzzle, and mostly it is. But the last moves on this board depend on something the numbers alone won’t tell you: every island has to end up in one network. Here’s a complete solve, with a reason for every bridge.
The rules in brief: bridges run straight across or straight down between islands. They can’t pass through an island or cross another bridge, and two islands can share one bridge or two. Each island needs exactly as many bridges as its number, and when you’re done you must be able to travel from any island to any other.
Starting board
Eight islands, no bridges yet
The islands sit on four rows and four columns, and small dots mark the empty spots. Rows count down from the top and columns across from the left. In later diagrams a soft glow marks the bridges added in that step, a ring marks an island with all its bridges, and an × marks two islands that can’t be joined. The × is just for these diagrams.
Before you read on, find the island with only one neighbour.
1. Start with an island that has one neighbour
From any island, look straight up, down, left and right, and stop at the first island in each direction. Those are its neighbours, the only islands it can ever connect to. The 1 in row 2, column 1 has nothing above it, nothing below it and nothing to its left. Its only neighbour is the 3 just to its right, so that’s where its one bridge goes.
One neighbour, one bridge
The 1 is complete. The 3 it joins now has one bridge and needs two more.
2. Give the 4 everything on offer
The 4 in row 1, column 2 sits on the top edge with nothing to its left, so it has just two neighbours: the 3 to its right and the 3 below it. Two islands can share at most two bridges, so two neighbours can give it four at most. It needs exactly four, which means a double bridge to each. A 4 in a corner always works this way, and so does a 6 with three neighbours.
Two doubles
Look for numbers as big as their neighbours allow. They settle several bridges at once.
3. Follow up on the islands you just touched
The 3 in row 2, column 2 now has two bridges from the 4 and one from the 1. That’s three, so it’s full, and it can’t connect down to the 2 below it.
The 3 in the top-right corner has two bridges from the 4 and needs one more. Its only other neighbour is the 2 in row 3, column 4, so a single bridge runs down to it. You could have made a start here without the 4, too: a 3 with two neighbours always needs at least one bridge to each, because one neighbour can supply two at most.
Full islands close their other connections
Both 3s are complete. The × shows that the 3 in row 2 and the 2 below it can’t be joined.
4. Draw the bridge you’re sure of
Now look at the 2 in row 3, column 2. Above it is the full 3, so that way is closed. To its right, the 2 in row 3, column 4 already has a bridge from above, so it can take only one more. The 2 needs two bridges and the right can supply one at most, so at least one has to go down to the 2 in row 4.
You don’t know yet whether it’s one bridge or two, but you can draw the one you’re sure of.
At least one bridge down
The new bridge might still become a double. The next step settles it.
5. Don’t cut off a pair
Could the two 2s in column 2 share a double bridge? Then each would have both its bridges, both from the other, and neither could connect to anything else. The pair would be sealed off from the rest of the board, and every island has to join one network. So the bridge between them stays single. Two 1s can never be joined to each other for the same reason.
That means the 2 in row 3, column 2 sends its second bridge to the right. The 2 in row 3, column 4 now has one bridge from above and one from the left, so it’s full and can’t connect down to the 1 below it.
A finished pair can’t stand alone
The counts would allow a double between the 2s. It’s the one-network rule that says no.
6. Finish with the last 1
The 1 in the bottom-right corner has two neighbours: the full 2 above it and the 2 in row 4, column 2 to its left. Only the left is still open, so its bridge goes there. That gives the 2 in row 4 its second bridge as well.
Solved
Every island has its number, no bridges cross, and you can travel from any island to any other along the bridges.
Questions for your next board
Every bridge here came from a handful of questions. Which island has the fewest neighbours? Which number is as big as its neighbours can give? After a bridge goes in, which islands just became full, and what does that close off? Where must at least one bridge go? And before you finish a pair or a small group, will it still connect to everything else?
This board was made for the lesson and checked for a single solution; it isn’t from the Daily set or the Weekly collection. When you’re ready, try Bridges in the Library, or read the Bridges guide for the rules and controls.