How to Solve Gokigen Puzzles

Gokigen Naname, commonly shortened to just Gokigen, is a line placement puzzle. Place a slanted line into each cell, following numbered clues at grid intersections. You may have seen it before under the name “Slant” or Slalom,” but Gokigen Naname is a Japanese puzzle introduced by Nikoli magazine. Interestingly, the name literally translates as “to be in a bad mood.” In doing my research for this puzzle, I discovered that “Gokigen Naname” is also the name of a Virtual YouTuber.

Rules
Basic Techniques to Solve
Solving the Puzzle

Rules

Draw one and only one diagonal line in each cell of the grid.
Numbers indicate how many lines intersect a circle.
Solution lines may not form any closed loops.

Pattern of lines around a 1 clue.

One of two options for lines around a 2.

The other pattern of lines around a 2.

Pattern of lines around a 3.

Pattern of lines around a 4.

Don’t create a closed loop of any size or shape.

Start Simple

As with all logic puzzles, begin by looking for the easy “break-in” clues. These starting lines provide the implied lines for other clues, allowing you to move forward in your solve. Examine the grid for the following patterns:

0 in the corner or along the edge: All lines will avoid the clue. A 0 will never appear in the middle of the grid, because it would create a loop around itself. Note that the border itself does not close any loops.
1 in the corner: This clue borders only one cell, so the line must connect to it.
2 along the edge: This clue borders exactly two cells, so both lines connect, forming a V-shape.
4 in the middle: Again, the number of bordering cells matches the clue, so all lines connect.

0s only appear on the border.

A 1 in the corner has only one cell, which must connect.

A 2 on the edge forms a V-shape.

As we saw before, a 4 clue forms an X.

Basic Implications

Once you have your starting lines, you can begin looking at clues near them to draw more slants. The simplest implications, of course, are when you have already completed a clue. When that happens, simply place lines in the remaining adjacent cells so they avoid the clue. A slightly more subtle case is when you’ve drawn a line that avoids a 3 clue. This implies that all remaining adjacent cells must connect to the 3. Below are a few patterns showing these completion-type implications. A pattern of lines in blue implies the green lines, and vice-versa.

Three lines avoiding the 1 means the last cell connects. Similarly, if you know the connection, the rest avoid it.

Similarly, if you know the line that avoids a 3 clue, you know the rest connect.

If two lines connect to or avoid the 2, you can infer the position of the remaining two lines.

Again, if you know two lines, you know the other two. This pattern comes in handy for chains of 2s on the same line.

The other type of basic implication is a bit more abstract. In these cases, we use a letter, such as A or B, to represent a pair of slants that must lean the same direction. We won’t know the direction initially, but once we determine one, the matching letter must be the same.

A 1 on the edge has two bordering cells that must match. Otherwise, you would break the clue with zero or two connections.

Similarly, with two established lines on one side of a 1, the opposite side must be matching slants. If not, you get a loop, or two connections.

If you have an existing pair of matching slants on one side of a 2, the opposite side must also be matching slants.

Note that they don’t have to match the first pair, only each other. You are avoiding a break of the 2 clue with three lines or only one.

Finally, if you already have two connected lines on one side of a 3, the opposite side must be a matching pair. If not, you have only two connections, or four.

Don’t Close Loops

As you solve, watch for potential loops, and you may be able to deduce some information about nearby cells. Some traps are easier to see than others; experience helps here. To get you started, here are a few basic loop patterns, and what they imply.

Gokigen Loop 3 Semicircle — First Example: A 3 facing a V-shape.

Gokigen 3 Semicircle Closed Loop — If we fill the two cells with opposite slants, we get either a closed loop, or only two connections to the 3.

Gokigen 3 Semicircle Implication — The slants facing the V must be identical. From this, we deduce that the opposite side of the 3 must both connect.

Gokigen 3 Large Loop — The same technique holds true for larger loops, if the final corner faces the 3 clue.

Gokigen 3 Larger Loop Implication — We can deduce exactly the same information about the cells around the 3 clue.

Gokigen 2 Semicircle Example — Second Example: 2 faces a V, with an avoidant external slant.

Gokigen 2 Closed Loop — If the other external slant isn’t the same, we force the remaining connections to close the loop.

Gokigen 2 Semicircle Matching External — The external slants must match, and we know this implies another matching pair on the other side of the 2.

Gokigen 2 Larger Closed Loop — Just like the 3 in the first example, the principle remains the same with larger loops.

Gokigen 2 Larger Loop Implication — We can come to the same conclusions if a loop’s corner faces the 2 with a slant avoiding the clue on the opposite side.

Gokigen 1-2-1 V-Shape — Third Example: 2 diagonally adjacent to two 1s, connected to one of them.

Gokigen 1-2-1 V Loop Closed — If we connect to the other 1, the lines you force will form a closed loop.

Gokigen 1-2-1 V Implied — We place the slant to avoid the second 1. Now we can apply the matched pair deduction.

Gokigen 1-2-1 Line Closed Loop — The same rule holds true for a 1-2-1 in a line. Connecting to the second 1 forces a closed loop.

Gokigen 1-2-1 Line Implication — Remember, this was based on a known connection. Our deduction shows the opposite slant avoids the second 1.

Gokigen 1-2-1 Line Border — The border never closes a loop, so you can’t make the same deduction if at least one of the 1s is on the edge.

Gokigen Parallel Line Loop — Fourth example: Common corners

These lines around the 1 imply a matched pair of slants.

Similarly, these lines also imply a matched pair, so all three cells have the same slant.

Notice that if the three parallel lines slant to the left, they close a large loop.

So all of them must slant the other direction. Watch out for implied parallel lines which can close a loop.

Basic Clue Patterns

Now that we’ve seen a few loop patterns, let’s look at some simple clue pairings. For brevity, “next to” means the second clue is one space away in the same row or column. Interestingly, almost all of these patterns create a matched pair of slants in the cells between the clues.

1 on the edge next to a 1 or 3: The inner cells must be a matched pair. Cells outside the 3 connect, while cells outside the second 1 are avoidant.
1 next to another 1: The cells between them must be a matched pair. The outside cells all avoid the 1s.
3 next to another 3: Inside cells must be a matched pair. The outside cells all connect to the 3s.
2 with an outer connected line, next to a 1: The cells between them must be a matched pair. All other outside lines avoid the clues.
2 with an outer avoidant lines, next to a 3: The inner cells must be a matched pair. All other outside lines must connect.
2 next to another 2, with an avoidant line in opposite outer “corners”: Inner cells must be a matched pair. All other outer cells connect. This is true for longer chains of 2s, but each matched slant pair does not need to be identical.
2 next to another 2, with connecting lines on the same outer “side”: The inside cells must be a matched pair. Other outer cells avoid the clues. Again, this also holds true for longer chains of 2s, if they have the required outside slants..
1-1 diagonal pair, neither on the border: Connecting them would force a closed loop, so the line between them must avoid both clues.

A 1 on the edge implies a matched pair of lines. On the other one, this implies these two lines avoid it.

By a similar set of implications, the lines on the other side of the 3 must connect.

Cells between a 1-1 must contain a matched slant pair, each connected to one clue. This prevents a clue from having two connections. So all outside cells are avoidant..

Similarly, cells between a 3-3 must contain a matched pair, to avoid a missing connection. Since each “A” will avoid one of the clues, the outside cells must all connect.

A line connected to a 2 facing a 1 needs the cells between them to be a matched pair or a clue would be overcrowded. Other outside cells must be avoidant.

A line avoiding a 2 facing a 3 needs the cells between them to be a matched pair or a clue would be undercrowded. Other outside cells must be connected.

Avoidant lines in opposite “corners” of a 2-2 pair force matched slants between them, and the other outer cells to connect.

This is also true for longer chains of 2-clues, but each matched slant pair does not need to be identical.

Two connected lines on the same outer “side” of a 2-2 pair means the other outer cell must match or overcrowd the other clue. That means a matched pair in the middle.

Again, this is true for longer chains of 2s, but each pair of slants need not be identical.

Finally, connecting a 1-1 diagonal pair forces a closed loop if neither 1 is on the border.

Therefore, a 1-1 diagonal pair in the middle of the puzzle always has an avoidant line between them.

Complex Clue Patterns

By now, you have enough information to solve our example puzzle, but I learned a few more complicated patterns that may give you additional insight. Draw them for yourself and experiment with different slants to see how effects can cascade. Here are the concepts demonstrated in the next set of diagrams:

We saw earlier that when you have two different matched pairs that intersect, they must all be the same slant.
A semicircle shape around a 1 acts mostly the same as though that 1 were on the edge. The remaining two cells must be a matched pair.
2 clues create a kind of chain effect, because the configuration on one side forces a similar configuration on the opposite side. A pair of avoidant slants implies a pair of connected lines, and vice-versa. A matched slant pair means another matched slant pair, even if it doesn’t necessarily slant the same direction.

1s on the border have matched slants. So, a chain of 1s all have the same slant, even around a corner.

A semicircle next to a 1 forces a matched pair. So if it faces a 3, it implies the outer cells connect.

Notice the intervening 2 still creates the same relationship between the 1 and 3.

No matter how many 2s you put between this 3-3 pair, the result is the same. This is because an unmatched pair on a 2 will give one of the 3s too few lines.

This last one looks complicated. We have a cross formed of 2s with a few known lines.

Think of it as separate chains that cross. If we draw an opposite slant at either end of this chain, we invalidate the 2 at the other end.

So we can infer these lines at the top and bottom. Now let’s look at the other chain.

Again, an opposite slant at one end invalidates the 2 at the other end.

So again we infer matching slants. Notice that at the intersection, we must have 4 identical slants, because multiple matched pair patterns intersect.

Solving the Puzzle

I know I just threw a lot of information at you. Don’t worry – you don’t need to memorize all the patterns to solve Gokigen puzzles. That was just to demonstrate some simple examples of the deductive reasoning enforced by the rules. Now let’s see them in practice.

First Steps

Let’s start with these 2s on the border. They create simple V-shapes.
That causes several implications, but we’ll start with these edge 1s. We know the slants in front of them match.
We already completed these 1s, so we can place all the avoidant lines around them.

Using the Edge 1 pattern, we know these lines must match the new slants.
We can draw the avoidant lines for completed 1s. While we’re at it, we can complete the corner 1s at the bottom.
Thanks to basic implications, we can go ahead and solve these clues.

Adding Simple Patterns

So far, we’ve focused on the edge and taken small solving steps. Before we move on, let’s look at the whole grid. By using the basic patterns, we can add quite a few more lines to work with.

We have a lot of diagonal 1 pairs, which means we place lines that do NOT connect them.
This pair of facing 3 clues creates an X-pattern in their outer cells.
Remember, the 2 provides a chain effect to patterns. So these 3s also form an X in their outer cells.

We can apply that same principle to these 1-1 pairs with a 2 between them. Outside cells contain avoidant lines.
Remember 2-3 with an outside avoidant line? That means a matching slant by the 2 line, and a partial X outside the 3.
Before we move on to the next phase, we have a bunch of simple clues we can solve.

Filling in Gaps

Look at all these matched pairs we’ve created with an existing slant. We can quickly fill in the identical cells.
Mark matched sets if you can. That way, solving one cell allows you to place chains of lines.
For instance, this V-shape translates forward through these 2s. This solves all these cells.

Similarly, solving this 1 fills both “C” cells.
Solving this 3 fills both of the “B” cells.
Completing this 2 gives us the avoidant line that solves the 3. That will fill in the “C” cells.

We can complete this 1. Plus, have a matched pair by the 3, which means the outer cells are both connected.
When we finish these 2s, we will also be able to complete the 1s and mark the last few cells.
We’ve solved the Gokigen puzzle!

Rules