The rules are simple:
- each cell has a diagonal line
- each number has exactly that many lines connected to it
- lines must not form a closed loop

This guide applies to all implementations of Gokigen Naname[en.wikipedia.org], aka Slant[www.chiark.greenend.org.uk] or Slants.

If you wanted to, you can just solve every cell in this version with the "Show Mistake" button and time. But if you want to "really" solve the puzzles, you can derive many other patterns from the base rules. This guide is a reference/explanation for those. "Before" pictures are on the left, "after" pictures are on the right.

I tried to use 1/2/3 when referring to clue-numbers and spell out the numbers otherwise ("one connection", "two outer cells", etc.)

The "inner cells" of a pair are the two cells between them (or you can think of them as being shared by the two clues). The "outer cells" of a pair are the other four cells.

There are some obvious things that I won't get pictures for:
- a 4 needs all four cells connected to it, a 3 with one cell not connected needs the other three connected to it, etc.
- a 0 can't have any cells connected to it, a 1 with one connection can't have any other connected cells, etc.

If two 1's are diagonally adjacent and neither is on the edge, there can't be a line connecting them. Unlike the other patterns, the explanation for this one is in a later section of this guide.


You can solve the four outer cells in this pair. Reasoning: pretend an outer cell was connected to a 1. This would mean that both inner cells of that 1 couldn't connect to it, but then both inner cells would have to connect to the other 1. This isn't allowed, so none of the outer cells can connect to either 1.


If one of the 1's is on the edge, you can use the same reasoning and just solve the outer cells of the 1 that's not on the edge.


A 3-3 is the opposite. Pretend that both inner cells were connected to one of the 3's. This would only leave two possible connections for the other 3. So each 3 can only have one inner connection, and their other two connections must be outer.


If both inner cells were connected to the 3, there wouldn't be any connections left for the 1. So the 3 must have two outer connections.


Similar reasoning as above. The 2 needs to have one outer connection. If you've already eliminated one of the options, you can solve the other outer cell.

If either outer cell of the 1 was connected, then both inner cells would be connected to the 3, putting it at four total connections.


This one outer cell lets us solve the other three. If both inner cells were connected to the 3, the 2 would only have one possible connection left. If both inner cells were connected to the 2, the 3 would only have two possible connections left.


If another outer cell of either 2 was connected, then both inner cells would be connected to the other 2. The other 2 would have too many connections. So neither 2 can have another outer connection.


If both inner cells were connected to a 2, then the other 2 wouldn't have enough possible connections left. So both 2's need a connection in their outer cells.


If the 2's other outer cell is connected, both inner cells would be connected to the 1.
If either of the 1's outer cells were connected, both inner cells would be connected to the 2, giving it three total connections.


Same reasoning as above, except you can only solve the 2's other outer cell.


In this example, if the 3 had 2 connections on the right side, it would create a closed loop. This isn't allowed, so the 3 must have 1 connection on the right side and 2 on the left. This is the smallest example, but remember that it applies to bigger loops as well.


Same reasoning as above, except you need to already know that one of the 2's non-loop-closing cells is not connected.

Let's talk about extensions. So far all of the patterns have been adjacent cells, but can we do anything over longer distances? Consider the following image.



We would like to use the 1-1 pattern, but there's a 2 sitting in the middle. It seems like we can't do anything, but actually:

Pretend the outer cell of a 1 on the left was connected. This means the cells between that 1 and the 2 are connected to the 2. Since that's both of the 2's connections, we can "solve" its other cells.



! Just like the normal reasoning for a 1-1 pair, the other 1 has too many connections. Just like the normal reasoning for a 1-1 pair, this means that the outer cells of the 1's can't be connected.



The 1-1 pattern has been applied as if the 2 in the middle didn't exist!


For some help understanding why this is possible, look at apocalyptech's comments.

Here are several examples:

1-3 on edge, extended

1-3 with both outer cells of 3 connected, extended

2-3 with an outer cell of 2 not connected, extended

2-2 with an outer cell of each 2 connected, extended

2-2 with an outer cell of each 2 not connected, extended

2-1 with an outer cell of 2 connected, extended

3 can't close a loop, extended

2 can't close a loop, extended

It's important to note that patterns extend across an unlimited number of 2's. Look at how this 2-3 pattern extends across all four 2's in the middle.

The first pattern in this guide was diagonally adjacent 1's, which can't be connected to each other. Look at what happens if they are:



The other lines around the pair are all forced, and they create a closed loop.

This is an example of a corollary of the "no loops" rule:
All paths must be connected to the edge of the grid.

If there's a path that's not connected to the edge, then the cells around them will form a closed loop. In our diagonal 1's example, the line between the 1's can't be connected to anything else because both 1's are "used up". Since it doesn't connect to the edge, there's a closed loop.

Here are other examples of WRONG connections that will create a closed loop.



These are just examples I made up, so they have other errors. Here's a real puzzle:



If you look at path that connects the 1 and the 2, you'll see that it doesn't touch the edge. If we block it in, it'll never be able to touch the edge and a loop will form. We need to extend the path.



In that example, it's relatively easy to see that a loop will form even if you don't look at the path. In the same puzzle, there's a larger loop that's a bit trickier to spot.



How should we fill in the last cell in the middle row? We can look at the same path that we used in the last example. I've marked it below:



Notice how it doesn't connect to the edge and there aren't any other places it could use to get there. It's blocked off everywhere else. The only option is our unsolved cell, so we have to extend the path.



You could realize that filling in the cell the other way would create a loop around that path, but it can often be hard to see large loops like that. Sometimes it's easier to look at which paths need to avoid being boxed in.

There are other more advanced deductions, but I'm not really familiar with them. These should be enough to solve all the puzzles in this game without making random guesses.