Solving Hashiwokakero

The rules of this puzzle are deceptively simple, but solving them can be quite tricky. Here's a recap of the rules:

• Each island must be connected to every other island in a network.

• All bridges must begin and end at an island, going only in a straight line.

• Bridges can be placed left/right and up/down, but not diagonally.

• Bridges cannot cross islands or other bridges.

• A pair of islands can be connected by one or two bridges.

• The number of bridges leaving an island must match the number on it.

Solving these puzzles is a matter of finding unique bridge placements that then eliminate other possible placements, forcing a gradual progress to completion.

Edge Islands

The most productive place to start is usually the edge islands as these are constrained to just 2 or 3 bridge directions.

Corners:
An island in a corner showing 3 must have at least one bridge in each possible direction. If it has a 4 then there must be two bridges in each possible direction.

Edges:
An island on an edge showing 5 must have at least one bridge in each possible direction. If it has a 6 then there must be two bridges in each possible direction.

Internal Islands

There are usually a number of starting points on the internal islands that can help eliminate a lot of possible placements.

8's:
Look for any island with an 8 showing as this must have 2 bridges in all four possible directions.

7's:
Look for any island with a 7 showing as this will have at least one bridge in all four possible directions.

Internal Corners and Edges:
Often a lack of islands to connect to for islands close to the edge will lead to an 'internal corner' or an 'internal edge'. The 4 on the left end of the second row down in the example grid shows an internal corner. There are no islands above or to the left and so bridges are constrained to two directions. Since the number is 4 there must be two bridges in each possible direction.

The 4 to the right end of the fourth row down is an internal edge. There are no islands to the right and so there must be at least one bridge in each possible direction from this island.

Once you begin to place bridges further internal corner and edge situations will occur as the bridges will prevent connection to islands in certain directions.

Isolators:
Placing a bridge that would cut off one group of islands from another and prevent any possible connection are not allowed.

Placing bridges that prevent further connection to the wider network by filling the bridge quota are not allowed. A simple example of this would be placing a bridge between two islands each with a 1 when there are other islands on the grid.

That's it. These techniques should get you to the solution of even the most difficult Hashiwokakero puzzle. Just work through the unique placements and eliminate possible placements as you go, creating new unique placements.