1. Draw a single loop that connects the [parts of the grid]
2. The loop may only travel horizontally or vertically, not diagonally (so all turns are right angles).
3. The loop may only turn at [parts of the grid]
4. The loop may not cross itself or branch off.
While all of the puzzles that use these rules have other rules as well, they all can use these similar theorems.
All examples here are a generic loop grid where they turn at the centers of the grid cells. Slitherlink, which instead just uses dots, still uses all of these same techniques and theorems, but just looks different.
Important Tip: write something (like a small x) anywhere you know the loop does NOT go.
This can be extremely useful depending on the puzzle. Also, the edges of the grid act like an x.
Theorem 1: If the loop goes in and out of a cell in two of its four directions, the loop may not go in the other two directions.
Think of this generic loop puzzle:
By this theorem, we can put x's on all of these spots:
Theorem 2: If an empty cell has an x on three sides, there is an x on the fourth side.
Our previous example becomes this (after multiple iterations of applying this theorem):
Theorem 3: If a line going somewhere creates a closed loop that does not include the rest of the lines, the line may not go there.
That makes this x go here:
Theorem 4: If a line goes into a cell with two x's, another line must be in the fourth direction of the cell.
That makes this happen:
Now, by Theorem 1, we have this x:
And by Theorem 4 again:
There's a few things you should note about how this puzzle was solved:
1. Most of it was placing x's. That's how it always is. Those x's can be very important.
2. Notice that it could have been done differently. We didn't have to use Theorem 3 and could have just used Theorem 4 to do it instead. I just wanted to show Theorem 3.
3. I didn't place an x between rows 2 and 3 in column 3. That was originally an accident, but I kept it. Sometimes you see the solution without the x's, and in that case, you don't need to do place an x there.
Theorem 5: There must be an even number of open loops going into a closed area.
This one might seem kind of weird, but an example should show what it means. Consider this example:
(I don't care that you can solve this with the other theorems, I'm just showing how this one is used)
We'll actually use this one twice to show the different ways to use it. First, consider the bottom right. There's only one way in, but if that open loop goes that way it gets stuck. The reason is that the bottom right is a closed area, and if that one loop went in it would have an odd number of open loops going in.
The other way is to think of the top left. It has three loops going into it (the three are the ends at R1C3, R2C2, and R3C1), while there another one that can if it wants to (R3C2). If the one that's uncertain doesn't go up there, one of them will choke, because there isn't an even number of loops going into the area. That one at R3C2 must into the area.
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