Hashi solving strategy

So you understand hashi. You know what the rules are, and you know what you’re supposed to do. (Note: if you do not understand hashi, please go read “Understanding Hashi” first!) But you sometimes find the puzzles really hard to solve. Well, read these tips to learn how the experts solve even the toughest ones.

Use logic, not guessing

You could try to solve a puzzle by guessing, just putting bridges in places that “feel” right. But if you guess, you will probably fail many times before finding the actual solution. And when you do find it, you will not feel the thrill of victory so much as relief that the ordeal has ended. However, if you use logic instead of guessing, you will solve every puzzle on your first try, and every solution will be a gratifying conquest.

Using logic means following this one simple maxim: Don’t place any bridge, unless you know it must be in that place!

Look for these patterns

There are many bridges you can place right away, just by spotting certain patterns in the arrangement of the islands. Let’s talk about some of them, starting with the easiest.

For example: if you see an island with the number ④ on it, you know it has to end up with four bridges, because that is what the number means. And if you notice that this ④ island has only two neighboring islands it can connect to, that means it must send two bridges to each neighbor! You know this even without knowing any other information. It doesn’t even matter what the numbers on the other islands are. You can just place those four bridges right away.

In the same way, if you see a ⑥ island that has only three neighbors, you can place all six of those bridges, two to each neighbor. And if you see an ⑧, you can place all eight of them.

If you see a ③ island with only two neighbors, you might not know enough to place all three bridges yet. But you certainly know enough to place two of them. Because it can’t send all three bridges to one neighbor, it must connect to both of them with at least one bridge apiece. You should place those two known bridges right away, and save the third, unkown, bridge for later. Every bridge you can place helps solve the puzzle.

Using the same reasoning, a ⑤ island with only three neighbors must connect to all three of them at least once. And a ⑦ island must connect to all four of its neighbors at least once.

So far, the patterns we have discussed are all really about simple addition, and counting by twos. But now let’s talk about a different kind of pattern.

Avoiding isolation

You know from the rules that the solution must involve connecting all the islands together, that no islands can be isolated. Well, this advice follows logically from that rule: A ① island can never have a bridge to another ① island! If a pair of ① islands were bridged to each other, then they could never connect in any way to the rest of the islands.

How about a pair of ② islands? Those can connect, right? Yes, they certainly can, and sometimes do. But never with two bridges. A ② island can never have a double bridge to another ② island!

That’s why, if you see a ① with two neighbors, but one of them is another ①, you can immediately place a bridge going to the other neighbor (the one that is not a ①).

And if you see a ② with two neighbors, but one of them is another ②, you can immediately place a bridge going to the other neighbor (the one that is not a ②).

We can make similar logical deductions in slightly more complex situations. For instance, suppose you see a ③ with three neighboring islands. And one of those neighbors is a ① and another is a ②. You don’t have enough information to place a bridge to either one of those. But you do have enough information to place a bridge to the remaining neighbor! Because if the ③ only connected to the ① and the ②, all three of those islands would be isolated from the rest of the puzzle. So it must connect to the other neighbor with at least one bridge.

And the same is true of a ④ island with three neighbors, if two of the neighbors happen to be ②. That ④ cannot send two bridges to both of the ② islands, because that would isolate all three of them. So it must send at least one bridge elsewhere.

There are many pleasant variants of these patterns, which I will leave as an exercise for the reader. For instance, if you see a ④ island with three neighbors, and its neighbors are ②, ②, and ②, do you know what bridges must be placed? Think about it.

Every bridge you place helps

Because the bridges can never cross, every bridge you place can reveal new patterns in the rest of the puzzle. For instance, an island that used to have four neighbors may suddenly have only three, because a bridge has blocked its view of one of them. So after every move, take a fresh look at every island that was affected.

Keep thinking

This is how hashi puzzles are solved.

• Look for a place that a bridge has to be.
• Whenever you know where a bridge has to be, put it there.
• Placing that bridge will reveal new places where a bridge has to be.
• Repeat until solved.

Now, go have fun!