![[Our Pink House]](../../template/masthead.gif)
|
Overview
A hashi puzzle consists of a partial grid of numbers, generally rectangular. The goal is to connect the numbers with straight lines following these rules:
Here is a sample unsolved hashi grid and the solution:
 |
 | |
| Fig. 1: Unsolved grid | Fig. 2: Solved grid |
For the rest of this article, "neighbor" will refer to other numbers that a given number can connect to, which will not necessarily be the numbers that are physically close by; lines can only go horizontally and vertically, so numbers not aligned in that way are irrelevant to each other. Circles are darkened to represent that all of their connections have been made (that is, that rule 3 has been satisfied for that number), while blue lines and circles indicate areas of discussion.
Strategy Summary
Basic Strategy
If an even number n can only be connected to n/2 neighbors, it must connect by two lines to each.
If an odd number n can only be connected to (n+1)/2 neighbors, it must connect by at least one line to each.
As connections are made, look for cases where possible connections have been reduced.
Avoiding closed subsystems
1-1 and 2=2
Other common configurations
Finishing the puzzle
Basic Strategy
If an even number n can only be connected to n/2 neighbors, it must connect by two lines to each.
If an 8 appears anywhere in the grid, it must be connected to each of its four neighbors by two lines. Likewise, the same is true if a 6 appears and has only three neighbors, a 4 appears with only two neighbors, or a 2 appears with only one neighbor.
In the example, the 6 has three neighbors, so we can start there:
 |
| Fig. 3: 6 has 3 neighbors |
If an odd number n can only be connected to (n+1)/2 neighbors, it must connect by at least one line to each.
A 7 can have at most four neighbors. Let's say it's connected to three of those neighbors by two lines. It will have to be connected to the fourth neighbor to use up the seven available lines. Therefore, a 7 will be connected to all of its neighbors (by at least one line). The same is true for a 5 with three neighbors, a 3 with two neighbors, and a 1 with one neighbor.
It is common for a 3 to appear in a corner, with only two neighbors. There are two such cases in this puzzle:
 |
| Fig. 4: 3 has 2 neighbors |
The 5 on the bottom row has three neighbors, so we can make some connections there as well:
 |
| Fig. 5: 5 has 3 neighbors |
Note that the numbers still don't have enough lines. A general strategy in hashi puzzles is to draw connections as required in order to eliminate possibilities; if done correctly, the final solution will follow the rules.
As connections are made, look for cases where possible connections have been reduced.
In the example, consider the 5 (on the bottom). It has one connection to the 1, which is the most the 1 allows. That leaves it with two other neighbors and four possible connections. Therefore, it must be connected by two lines to its other neighbors:
 |
| Fig. 6: 5 has a neighbor of 1 |
Likewise, the 2 in the upper right corner can have at most one connection to the 1 below it, so it must have a connection to the 2 next to it.
 |
| Fig. 7: 2 has a neighbor of 1 |
Avoiding closed subsystems
1-1 and 2=2
1-1: One important key to solving hashi puzzles is the fourth rule. For instance, if there are two 1s next to each other, they cannot cannot to each other because there would then be no way for them to connect to any of the other numbers; they would create a closed subsystem.
2=2: Likewise, if there are two 2s next to each other, they can share at most one connection; the other connection has to go elsewhere. In the example, this is true of the 2s in the upper right and lower right corners; each can be connected to exactly two other numbers, one of which is a 2. Therefore, we can draw in lines thus:
 |
| Fig. 8: 2 and 2 are neighbors |
Other common configurations
1-3=2: Another common configuration is a 3 next to a 1, a 2, and one other number. In that situation, if the 3 connects twice to 2 and once to 1, a closed subsystem is created. Therefore, the 3 has to share at least one connection with its third neighbor.
1-2-2-1: If there are two 2s surrounded by two 1s, at least one of the connections on each 2 has to be shared with some other neighbor.
2=4=2: If there is a 4 with two 2s and one other neighbor, it has to share at least one connection with the other neighbor.
These are just some of the simpler and more common configurations. There are many others that result in closed systems, and therefore are avoided.
Finishing the puzzle
Using these strategies iteratively, we can now solve the puzzle. In the next few diagrams, some connections are shown which can be deduced from the previous state of the grid.
 |
 | |
| Fig. 9: Iteration 1 | Fig. 10: Iteration 2 |
 |
 | |
| Fig. 11: Iteration 3 | Fig. 12: After Iteration 3 |
At this point, we're left with a more complex application of the closed subsystem prohibition. There are two ways to connect the 4 to its neighbors: It can either double-connect to the 2 to the right, or it can connect to the 3 below it. However, if we connect it to the 2, we wind up with all numbers satisfying the first three rules, but two large disconnect chunks of grid (colored red and green here) rather than one connected solution:
 |
| Fig. 12: Red and green disconnected |
Therefore, we know that 4 is connected to the 3 below it, and we can finish the puzzle:
|
| Fig. 13: Completed puzzle |
Conclusion
These strategies should be sufficient to get deep into most any hitori puzzles you choose to tackle. As you solve more puzzles, you will begin to see other patterns that will simplify solving. One key to more advanced hashi puzzles is trial and error: Try making a connection and see what other connections have to be made; incorrect connections will lead to impossible situations.