Slitherlink

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Moderately difficult Slitherlink puzzle (solution)

Slitherlink (also known as Fences, Takegaki, Loop the Loop, Loopy, Ouroboros, Suriza and Dotty Dilemma) is a logic puzzle published by Nikoli. As of 2005, 17 books consisting entirely of Slitherlink puzzles have been published by Nikoli.

1 Rules
2 Solution methods
3 History
4 Videogames
5 See also
6 References
7 External links

[edit] Rules

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 single 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 shapes can be used in lieu of the standard grid, as long as each tile has 4 sides. These include snowflake, penrose, laves and altair. They add complexity by increasing the possible paths from an intersection, but the same rules apply to their solution.

[edit] Solution methods

Whenever the number of lines around a cell matches the number in the cell, the other potential lines can be eliminated. This is usually done with an X.

A key to many deductions in Slitherlink is that every point has either exactly two lines connected to it, or no lines. For example:

A 1 in a corner.

If a 1 is in a corner, the actual corner's lines may be X'ed out, because a line that entered said corner could not leave it except by passing by the 1 again. This also applies if two lines leading into the 1-box at the same corner are X'ed out.

A 3 in a corner.

If a 3 is in a corner, the two outside edges of that box can be filled in because otherwise the rule above would have to be broken.

A 2 in a corner.

If a 2 is in a corner, two lines must be going away from the 2 at the border.

A 3 adjacent in a 0.

If a 3 is adjacent to a 0, either horizontally or vertically, then all edges of that 3 can be filled except for the one touching the 0. In addition, the two lines perpendicular to the adjacent boxes can be filled.

Two adjacent 3s.

If two 3s are adjacent to each other horizontally or vertically, their common edge must be filled in, because the only other option is a closed oval that is impossible to connect to any other line. Second, the two outer lines of the group (parallel to the common line) must be filled in. Thirdly, the line through the 3s will always wrap around in an "S" shape. Therefore, the line between the 3s cannot continue in a straight line, and those sides which are in a straight line from the middle line can be X'd out.

Two diagonal 3s.

Diagonal 3s with 2 between them.

One or more diagonal 2s terminated by two filled lines pointing into them.

Diagonal 3s with 2 between them.

If two 3s are adjacent diagonally, the edges which do not run into the common point must be filled in.
Similarly, if two 3s are in the same diagonal, but separated by any number of 2s (and only 2s) the outside edges of the 3s must be filled in, just as if they were adjacent diagonally.

A 3 next to a line.

Also, If a 3 is adjacent to a Zero Diagonally, both sides of the 3 that meet the zero's corner must be filled. This is because if either of those sides were open the line ending in the corner of the zero would have no place to go.
Similarly, If a 3's corner has X's in both directions going away from that same corner, then both sides of the 3 that meet that corner are filled. This is because if either of these sides of the 3 were open, the line coming from the 3 could not go in either of those X'd directions and it also couldn't close the 3 on itself. Therefore that side of the 3 can not be open and must be filled.

If a line reaches a corner of a 3, there must be lines on both sides of the 3 that said corner is not adjacent to, because if the 3's sole empty space were not adjacent to it, the corner would have three lines connected to it. Furthermore, the segment leading away from the 3 at the corner reached by the line must be empty; if it were filled, neither of the remaining 2 undetermined sides of the 3 would be able to contain a line.

A 2 next to a line.

When a 2 has a line adjacent to it, and one side that is X'd out, then a line comes into the 2 from the opposite corner where the X and the filled line meet. This is because there must be one and only one other side filled in around the 2. If the line wrapped around that corner, there would be too many sides filled in, so the line must exit the 2 at that corner. The inverse of this is, if a line comes into a 2 at a corner, and one of the sides not adjacent to that corner is X'd out, then the side adjacent to that X that is not adjacent to the corner where the line comes in must be filled in.

A 1 next to a line.

If a line comes into a corner of a 1 and if of the three remaining directions that the line can continue, the one that is not a side of the 1 is a known blank, then the two sides of the 1 opposite that corner can be X'd out.

In a closed-off region of the lattice (from which there is no path for any lines to "escape"), there cannot exist an odd number of unconnected segment-ends, since all of the segment-ends must connect to something. Often, this will rule out one or more otherwise feasible options.

In an exceptionally difficult puzzle, one may use another mathematical theorem, which states that any open curve that starts and ends outside of a closed curve must intersect the closed curve an even number of times. In particular, this means that any row of vertical lines or any column of horizontal lines must have an even number of lines. When only one potential line segment in one of these groups is unknown, you can determine whether it is part of the loop or not with this theorem.

A simple strategy to assist in using this theorem is to "paint" (sometimes called "shade") the outside and the inside areas. When you see two outside cells, or two inside cells be next to each other then you know that there is not a line between them.

[edit] History

Slitherlink is an original puzzle of Nikoli; it first appeared in Puzzle Communication Nikoli #26 (June 1989). The editor combined two original puzzles contributed there. At first, every square contained a number.

[edit] Videogames

Slitherlink video games have been featured for the Nintendo DS handheld game console, with Hudson Soft releasing Puzzle Series Vol. 5: Slitherlink in Japan on November 16, 2006, and Agetec including Slitherlink in its Nikoli puzzle compilation, Brain Buster Puzzle Pak, released in North America on June 17, 2007.^[1]