Hashi techniques

Hashi is a counting puzzle. Each island can reach only a few neighbours, and each pair can share at most two bridges, so comparing an island’s number with how much its neighbours could take decides a lot. Start with the big numbers in tight spots.

In each picture, the cells the technique decides are outlined in blue.

1. Full islands: every bridge doubled

   An island with two neighbours can take at most four bridges, so a 4 with only two neighbours needs a double bridge to each. The same goes for a 6 with three neighbours and an 8 with four.

2. One short of full: at least one each way

   A 3 with two neighbours is one short of the most it could take. Even if one neighbour got a double bridge, the other still needs one, so put a single bridge to each. A 5 with three neighbours and a 7 with four work the same way.

3. Don’t join two 1s

   A bridge between two 1s would finish both islands and cut them off from everyone else. Every island has to join one group, so the 1 in the corner can’t use its neighbour on the right. Its bridge must go down.

4. Don’t close off a pair of 2s

   A double bridge between two 2s would leave them as a group of their own. So the 2 in the corner can send at most one bridge right, and the other must go down to the 1.

When the counting runs out, think about the whole map: which groups are already joined, and which bridge is the last way out for one of them? If a bridge would finish a group of islands that still isn’t the whole board, it can’t be there.