Bridges

Start with the islands that can't share

The fastest way in is to hunt for islands whose number forces every bridge. An island showing 4 with only two neighbours has to run a double to each: there's no other way to reach four. A 3 tucked in a corner, with just two directions open to it, must send at least one bridge each way and then a second somewhere. Because any pair of islands can hold at most two bridges, a high number squeezed against few neighbours writes its own answer. Fill those in first, then treat each finished connection as a wall: bridges never cross, so a line you've committed to blocks anything trying to pass through it.

It helps to know the forced patterns by heart. An 8 anywhere must run doubles in all four directions. A 6 with only three neighbours is the same story: two, two, and two. A 7 always sends at least one bridge each way, because even doubles to three neighbours only add up to six. The arithmetic works smaller too: a 5 with three neighbours needs at least one bridge to each, and a 2 with a single neighbour is a forced double. None of this is guessing. It is just asking what the island could possibly reach if one of its directions stayed empty, and noticing the answer is not enough.

Keep the network in one piece

The rule people forget is the last one: every island has to end up joined into a single connected group. You can satisfy every island's count and still lose if you've built two separate clusters that never touch. So as the grid fills, watch for a region about to seal itself off. When you're stuck, find the island that's one bridge short and ask which neighbour still has room.

• Mark a pair as full once it has its two bridges, so you stop reconsidering it.
• Two 1-islands sitting next to each other usually can't be joined to one another. That would satisfy both but seal them off from the rest of the board.
• The same trap scales up: joining two 2-islands with a double bridge fills both and cuts the pair adrift, so unless they are the whole puzzle, at most one bridge can run between them.
• Count the free ends around a low-number island; often only one arrangement avoids over-connecting a neighbour.

When you stall

Every stall breaks with one of three checks. First, recount the islands around your last few bridges. A line you just placed may have quietly filled a neighbour and forced the rest of its count somewhere else. Second, run the one-short check: pick an island still missing exactly one bridge and list which neighbours actually have room, once full pairs and blocked lines are ruled out. Very often a single candidate is left. Third, use connection itself as a clue. If a corner of the board can only reach the rest of the network through one remaining line, that bridge must exist no matter what the numbers say, because the finished puzzle has to be one connected group. This third check is the one people forget, and on hard grids it is often the only way forward. Here is the second check in action: picture a 2 in the corner already joined by a single bridge to a neighbour that is now full. Its second bridge has exactly one other place to go, the island on its remaining side, so draw it without another thought. Work the three checks in order and you will never need to guess.

Hashiwokakero, straight from Nikoli

This puzzle goes by Hashiwokakero, Japanese for build bridges, and it comes from Nikoli, the Tokyo puzzle house that made Sudoku a household name. Nikoli first printed it in 1990, and it has been a fixture of their magazines ever since. You'll also see it sold simply as Bridges or Hashi, and the little numbered circles really do look like a ferry map of some crowded archipelago. The appeal is that it looks like a network-drawing doodle but plays like pure deduction: there's exactly one solution, and you never have to guess to find it. Once the numbers-and-lines logic clicks, a big grid comes apart in a very satisfying chain.