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Slitherlink


Slitherlink (also known as Fences, Takegaki, Loop the Loop, Loopy, Ouroboros, Suriza and Dotty Dilemma) is a logic puzzle developed by publisher Nikoli.

Slitherlink is played on a rectangular lattice of dots. Some of the squares formed by the dots have numbers inside them. The objective is to connect horizontally and vertically adjacent dots so that the lines form a simple loop with no loose ends. In addition, the number inside a square represents how many of its four sides are segments in the loop.

Other types of planar graphs can be used in lieu of the standard grid, with varying numbers of edges per vertex or vertices per polygon. These patterns include snowflake, Penrose, Laves and Altair tilings. These add complexity by varying the number of possible paths from an intersection, and/or the number of sides to each polygon; but similar rules apply to their solution.

Whenever the number of lines around a cell matches the number in the cell, the other potential lines must be eliminated. This is usually indicated by marking an X on lines known to be empty.

Another useful notation when solving Slitherlink is a ninety degree arc between two adjacent lines, to indicate that exactly one of the two must be filled. A related notation is a double arc between adjacent lines, indicating that both or neither of the two must be filled. These notations are not necessary to the solution, but can be helpful in deriving it.

Many of the methods below can be broken down into two simpler steps by use of arc notation.

A key to many deductions in Slitherlink is that every point has either exactly two lines connected to it, or no lines. So if a point which is in the centre of the grid, not at an edge or corner, has three incoming lines which are X'd out, the fourth must also be X'd out. This is because the point cannot have just one line - it has no exit route from that point. Similarly, if a point on the edge of the grid, not at a corner, has two incoming lines which are X'd out, the third must also be X'd out. And if a corner of the grid has one incoming line which is X'd out, the other must also be X'd out.

Application of this simple rule leads to increasingly complex deductions. Recognition of these simple patterns will help greatly in solving Slitherlink puzzles.

If a 2 has any surrounding line X’d, then a line coming into either of the two corners not adjacent to the X’d out line cannot immediately exit at right angles away from the 2, as then two lines around the 2 would be impossible, and can therefore be X’d. This means that the incoming line must continue on one side of the 2 or the other. This in turn means that the second line of the 2 must be on the only remaining free side, adjacent to the originally X’d line, so that can be filled in.
Conversely, if a 2 has a line on one side, and an adjacent X’d out line, then the second line must be in one of the two remaining sides, and exit from the opposite corner (in either direction). If either of those two exits is X’d out, then it must take the other route.


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