Advanced Hashi Techniques for Larger Boards

Niels Glodny ·

On a small board, the forced moves from How to solve Hashi usually carry you to the end. On a 25×25 board they run out, and you are left with islands that seem to have several options. The techniques below are about squeezing more out of each island and thinking about groups of islands instead of single ones.

Count capacity in each direction

Instead of counting neighbours, count how many more bridges each direction can still take. A direction’s capacity is the smaller of two numbers: two minus the bridges already there, and what the neighbour still needs. A direction blocked by a crossing bridge has no capacity.

Then compare. If an island still needs n bridges and the other directions can take at most c together, this direction needs at least n − c.

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The 1 above the 5 can take only one bridge. Left and right can take two each, which makes exactly five, so every bridge of the 5 is forced.

This one rule covers the beginner patterns as special cases. A 3 with two neighbours needs three, the other direction can take two, so this direction needs at least one. It also covers cases those patterns miss, like a 4 with three neighbours where two of them are 1s.

Keep groups open

As bridges go down, islands join into groups. A group that is not the whole board needs at least one bridge to the outside. Two things follow.

If a move would fill every island in a group, and the group is not the whole board, the move is wrong. Two 1s joined to each other are the smallest case. The same thing happens with three or more islands.

If a group has only one possible bridge left to the outside, that bridge is forced.

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The 2 on the right has one neighbour, so the double is forced. The 3 needs one more bridge, left or down.
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Left would close a group of three full islands, so the bridge goes down. The rest follows.

On large boards this matters most near the end, when the remaining islands sit in separate clusters. Before you finish a cluster, check that it still has a way out.

Check what a bridge crosses

A bridge removes every connection that would cross it. After placing a long bridge, look at the islands on both sides of it. Some of them have just lost a neighbour, which can make them forced.

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The 2 on the left edge has one neighbour, so its double bridge is forced. It runs between the 2 at the top and the 1 below it.
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The top 2 has lost its neighbour below, so both its bridges go to the 3. The 1 can only go right.

This works in reverse too. If an island must send a bridge in some direction, every possible bridge crossing that path is gone, even if you have not placed the bridge yet.

Use the fact that there is one solution

Every puzzle in this game has exactly one solution. If you find a part of the board that you could finish in two different ways, and nothing elsewhere decides between them, you have missed something. Usually it is the connectivity rule: one of the two ways closes off a group.

When nothing is forced, follow one assumption

Pick an island with few options, assume one of them, and follow the forced moves it leads to. If you reach a contradiction, that option is wrong. A contradiction is an island that can no longer reach its number, a group that gets closed off, or two bridges that would have to cross.

On the board below, counting and the group rule get no further. The 1 in the middle can still go up, left or down.

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Nothing is forced. The 1 has three options.
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Assume the 1 goes down. The 2 below it then takes only one bridge from the 5, so the 5 needs two from the 3 above it. These six islands are now full and joined only to each other.

That closed group is a contradiction, so the 1 does not go down. With that settled, counting and the group rule finish the board.

Keep the chain short. If it goes on for many moves without a contradiction, you have learned nothing, and you have to take every bridge of the chain back off the board. Try an island with fewer options first.

Working through a large board

  • Go over the whole board in passes rather than chasing one area. A move in one corner often unlocks something far away.
  • In the game, full islands turn green. Skip them when you scan.
  • Edges and corners tend to resolve early because islands there have fewer neighbours. The middle often comes last.
  • Late in the puzzle, check connectivity more often. Most mistakes on large boards are groups that get closed off without anyone noticing.

If a mistake has crept in, the hint button in the game points at a wrong or missing bridge.

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