Terms

From now on I'd like to use terms defined here. I have chosen terms used in solid modeling, which is used for computer aided design (including computer graphics and 3D geometry processing) for industrial products like automobiles. In fact what is required to solve this puzzle is very similar to what is required for automated product design using computers --- the leading-edge field. So, when you learn to understand this puzzle it enables you to read essays of that field --- of course I'm kidding.

Vertex

Dots in question. You need to connect two dots (vertically/holizontally) to make a line. I call these dots "vertices" here.

Edge

The line made by connecting two vertices vertically or holizontally. It may or may not have actual line.

Face

A square area formed with four vertices and four edges. Faces may have one number (0 to 3) in the center.

Size

Shows how large a question is by showing how many faces it has across and downward. Basically, questions are provided in square area.

Possible patterns for drawing lines

This section describes every possibility to draw lines around a face that has a number.

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These are all patterns for drawing lines around numbers.

Patterns of corner and end of line

There are only two cases with a vertex and loop: the loop goes or does not go through the vertex.

Usually, a vertex has neighboring vertices; each placed at its right, left, above, and below.

Wrong case Vertices never become the ends of loop. A loop cannot be formed if line ends at a vertex like this figure.

So, solving patterns for a vertex are limited to following cases. The last example shows a case when no line crosses the vertex.

Questions are always provided in square area (5x5 in "Rule details" section). This means there are exeptions with the outmost sides of the area:

You can draw lines only in two ways for the corner vertices. Since no edge can exist outside of this square, mark "x" on those edges.

If the loop goes through this vertex, then lines should be drawn on both edges.
If the loop doesn't go through it, then no line should be drawn on these edges and marked with "x".

Each of other vertices on the periphery has 3 neighboring vertices. There is no edge to extend outside of this square area, so mark "x" there.

So there are these cases for this kind of vertices. The last example shows a case when the loop doesn't crosses the vertex.

This explanation shows every basic technic and I can guarantee you that you will understand the rule fine.

If you can't wait to try on your own, go ahead. You can always come back to this section when you get stuck.