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Slitherlink (also known as Fences, Takegaki, Loop the Loop, Loopy, Ouroboros, Suriza, Rundweg and Dotty Dilemma) is a logic puzzle developed by publisher Nikoli.
| | This article needs additional citations for verification .(May 2011) |
Slitherlink (also known as Fences, Takegaki, Loop the Loop, Loopy, Ouroboros, Suriza, Rundweg 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.
 | This section contains instructions, advice, or how-to content .(February 2013) |
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.
If a region of the lattice is closed-off (such that no lines can "escape"), and is not empty, there must be a non-zero, even number of lines entering the region that begin outside the region. (An odd number of lines entering implies an odd number of segment ends inside the region, which makes it impossible for all the segment ends to connect. If there are no such lines, the lines inside the region cannot connect with the lines outside, making a solution impossible.) Often, this rule will eliminate one or more otherwise feasible options.
In the figure below, the line at the top-left will close off the top-right region of the lattice whether it proceeds down or to the right. The line to the right (around two sides of the 3) has entered the closed region. To satisfy the rule, the first line must enter the region, and the second line must not enter the region a second time. (Since the boundary of any closed region also closes off the remainder of the puzzle, the rule can also be applied to the larger, bottom-left region. To apply the rule, it is only necessary to count the lines crossing the boundary.)
In an exceptionally difficult puzzle, one may use the Jordan curve 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 the grid must have an even number of vertical lines and any column must have an even number of horizontal 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. This also means that if you mentally trace an arbitrary path from an outer edge of the grid, to another outer edge of the grid, the path will intersect the closed curve an even number of times.
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 next to each other, then you know that there is not a line between them. The converse is also true: if you know there is no line between two cells, then those cells must be the same "color" (both inside or both outside). Similarly, if an outside cell and an inside cell are adjacent, you know there must be a filled line between them; and again the converse is true.
In the figure below, if a solution could pass through the top and right sides of the 2, then there must be another solution which is exactly the same except that it passes through the bottom and left sides of the 2, because the squares to the top and right of the 2 are unconstrained (do not contain numbers). Also, the solution must pass through the top-right corner of the 2, otherwise there must be another solution which is exactly the same except that it passes through the top and right sides of the 2.
If there is a 2 in a corner, and the two non-diagonally adjacent squares are unconstrained, lines can be drawn as shown below. (In the figure, the question mark represents any number or blank, but the number will only be a 2 or 3. A puzzle with only one solution cannot have a 2 in a corner with two non-diagonally adjacent, unconstrained squares, and a diagonally adjacent 0 or 1.)
In the figure below, the circled points can be connected by a line directly between them, and also by a line that traverses the other three sides of the square that extends to the left of the points. It should be clear (with the red line ignored) that for both paths the remainder of the solution can be the same – since the constraints for the remainder of the solution are the same – so both paths are ruled out.
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 and the edges did not have to form a loop.
Slitherlink puzzles have been featured in video games on several platforms. A game titled Slither Link was published in Japan by Bandai for the Wonderswan portable console in 2000. [1] Slitherlink puzzles were included alongside Sudoku and Nonogram puzzles in the Loppi Puzzle Magazine: Kangaeru Puzzle series of games from Success for the Game Boy Nintendo Power cartridge in 2001. [2] Slitherlink games were also 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. [3]
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