How to Solve Hashi Puzzles
Hashi is short for Hashiwokakero, which literally translates to “build bridges.” Puzzle Communication Nikoli, a Japanese puzzle magazine, first published Hashiwokakero in September 1990, as well as earlier iteration in December 1989. Its inventor designed puzzles under the pen name “Lenin.” Interestingly, this was not to honor the Russian revolutionary, but simply a romanization of the designer’s name written in Chinese characters. Originally fascinated by atomic bonding diagrams, he wanted to design a puzzle with using double and single bonds, but he eventually changed it to the bridge-building theme we know today.
Rules
Connect all of the numbered islands into a single group, using a series of bridges.
- Each bridge is a straight horizontal or vertical line between 2 islands.
- Bridges may not cross other bridges or islands.
- Each island pair at most 2 bridges between them.
- Numbers indicate the total number of bridges connected to each island.
- You may not isolate any set of islands from the rest. All islands must form a single connected group. Think of it as being able to “walk” from one island to any other island.
Basic Techniques to Solve
- Divide by 2.
- Think of some edges as “corners.”
- Avoid creating isolated island groups.
- Look for patterns in groups of islands.
- Pay attention to new corners and edges you create.
Divide by 2
The most useful rule to remember is that no two islands are connected by more than 2 bridges. Find your starting point by looking at islands with a number of possible directions that’s about half of the clue. For example:
- 4 in the corner = 2 directions with 2 bridges each.
- 6 on the edge = 3 directions with 2 bridges each.
- 8 in the middle = 4 directions with 2 bridges each.
But that’s not all! If the island in these positions is only 1 less than the number that would completely fill all the directions, you know that only 1 bridge is missing from the maximum possible. You won’t initially know which direction the missing bridge is, but you can be certain that there is at least 1 bridge in each direction and a second bridge in all but one of them.
- 3 in the corner = 2 directions with 1 bridge each + 1 unknown
- 5 on the edge = 3 directions with 1 bridge each + 2 unknown
- 7 in the middle = 4 directions with 1 bridge each + 3 unknown
Edges as Corners
This tip isn’t always obvious to new solvers, but remember that a corner is simply an island with 2 options for a bridge. It doesn’t literally need to be in the corner of the grid. Sometimes you will have an edge island that form a straight line with two other edge islands, no third option. In such cases, you can treat the clue of the middle island as though it is in a corner.
This 3 only has two bridge directions, just like a corner 3.
Avoid Isolating Islands
Look for groups of islands that are easily cut off from the rest, because this will give you valuable information. All of the islands in the solution must be connected, so any bridge placement that isolates an island or cluster of islands can’t be true. Here are a couple of common examples.
Island Group Patterns
In more complex puzzles, you can look for patterns in island groupings. There are several which allow for some level of deduction to place a few bridges. Every known bridge cuts off potential crossings, and more placements reveal themselves. Here are a few groups to look for, followed by images showing why they work. Keep in mind that islands won’t always be as close as they are in the illustrations – the blank space between islands can be any distance, as long as it is unobstructed.
- 6 surrounded by four islands, one of which is a 1: There must be at least one bridge to each of the non-1 islands.
- 3 on the edge surrounded by a pair of 2s and a 1: There must be at least one bridge connected to each 2.
- 2 in the corner surrounded by 2s: There must be one bridge connected to each 2.
- 4 on the edge surrounded by 2s: There must be one bridge connected to each 2.
- 3 surrounded by four islands, three of which are 1s: There must be at least 1 bridge to the non-1 island.
New Corners and Edges
As you work through the puzzle, you will be creating new corners and edges with each bridge you place. Remember, bridges may not cross. Use this restriction to apply the other techniques to islands in the middle of the puzzle. You will see this technique in use as we solve the example puzzle.
Solving the Puzzle
Now we can apply these techniques to a full puzzle to see how they work together.
Examining the Border
Start with the border. These 3s each have only two directions, so we can place one bridge in each direction. 
We can solve this 6, and place two more bridges around this 5 on the edge. 
Notice that we’ve nearly closed off these two islands. We can only place their bridges in one direction.
Look back at these 3s. They each have two islands to bridge to, and one is already solved. So their final bridge must go the other way. 
This 6 has four bridges left to place, and only two directions, so we can solve it. 
This 1 has only one direction to place its bridge.
Working Inward
Each of these 3s face only 2 islands, so we can place one bridge in each direction. The 7 has three bridges and two directions left, so the principle is the same. 
We blocked in this 1, so we must connect it to the 2. The 2 then can only connect to the 4 with its second bridge. 
Each of these 2s has only 1 direction left for their second bridge. In both cases, they will also complete the 3 islands.
Speaking of completing 3s, each of these has only one direction left for their last bridge. 
Now we can complete the 7-4 pair with a single bridge, and the 2-5 pair with a double bridge. 
We’ve solved the puzzle! Note that every single island is connected in one group.
