Counting bridges, island by island
Hashiwokakero is a Japanese puzzle whose name means "build bridges", and many players simply call it Hashi or Bridges. The whole puzzle rests on one piece of arithmetic: an island can hold at most two bridges in each direction, and it can only reach as many directions as it has neighbours in line with it. Comparing an island's number with that maximum is where almost every deduction starts.
An island with four neighbours can take up to eight bridges. One with three neighbours can take six, two neighbours four, and a single neighbour two. Keep those maximums in mind.
Forced bridges first
Begin with islands whose number equals the maximum, or comes close.
- Full islands. An 8 always has four neighbours and needs double bridges to all of them. A 6 with three neighbours, a 4 with two, or a 2 with one are just as forced: draw two bridges in every direction.
- One short of full. A 7 with four neighbours, a 5 with three or a 3 with two must still have at least one bridge in every direction. Even if you cannot yet tell where the double goes, draw a single to each neighbour.
- Lonely islands. An island with only one neighbour in reach must send all its bridges that way, whatever its number.
Here is a worked example on the 7×7 Easy board. Say a 3 sits in the bottom-left corner. It can only look up its column and right along its row, so it has two neighbours and a maximum of four. Since 3 is one short of 4, it must have at least one bridge each way. Draw both singles. Now suppose the island directly above it is a 1. That 1 is already satisfied by the bridge you just drew, so it turns green. The corner 3 needs one more bridge, and it can no longer go up, so the second bridge goes right. One island just decided two of its neighbours.
Thinking about the whole network
Counting alone rarely finishes a board. The rule that every island joins one network does real work too.
- Never close off a small group. Two 1s side by side cannot be joined to each other, because that pair would be finished and cut off from everything else. The same goes for a 2 next to a 2 with a double bridge between them. On larger boards, look for any cluster whose numbers would all be used up inside itself.
- Bridges block lines. A bridge across the water stops any other bridge from crossing it. After placing a long bridge, re-check the islands on either side: some of them may have just lost a neighbour, which lowers their maximum and can force new bridges.
- Track what is left. For a half-finished island, subtract the bridges already drawn from its number and compare what remains with the directions still open. The same full and one-short checks apply again.
Common slips
- Drawing the obvious single and stopping. When an island is one short of full, players often draw one bridge and move on, forgetting that the remaining count may force a double later. Come back to it after each change nearby.
- Ignoring the red island. An island turns red when it has too many bridges. Fix that at once with Undo rather than carrying on, because every later move built on it is suspect.
- Joining the last two pieces too early. Near the end, check that the network will still connect before you finish an island that sits between two groups.
A path through the four sizes
Easy is 7×7 with about eight islands, small enough to solve with the full and one-short checks alone. Medium is 9×9 with around fourteen islands, where crossing bridges start to block options. Hard, at 13×13 with about twenty-four islands, is where the connection rule becomes essential. Expert is 17×17 with about forty, and rewards a steady routine: sweep for forced bridges, then check isolation, then sweep again.
Every board has exactly one solution, so a hint always moves you toward it. If you use one, look at the island it touched and work out which rule would have found it. If you like this blend of counting and connection, Slitherlink at /games/slitherlink asks you to keep a single loop joined in a similar way.
