Numberlink

Numberlink is a type of path-drawing logic puzzle, in which solvers must connect pairs of identical numbers within a grid via lines, while avoiding crossing other connections.

Background
Numberlink went through a few variations before being picked up and popularized by Nikoli. The first was a modified version involving connecting houses with gates within a large park, developed by Sam Loyd for an 1897 edition of the Brooklyn Daily Eagle. Later, in 1917, a closer version showed up in Amusements in mathematics by Henry Ernest Dudeney, who replicated something very close to today's Numberlinks but utilizing letter pairs instead. When the genre did make its way to Nikoli, it showed up in two forms: Arukone, which used letter pairs like Dudeney's puzzle, and Nanbarinku, which used number pairs.

Puzzle Application
Traditionally, Numberlinks are made up of a series of numbers scattered around an otherwise blank grid. Each number should have an identical number (only one) somewhere in the grid. The goal is then to connect every pair of numbers by travelling orthogonally through cells so that none of the connecting lines cross, and so that all cells of the grid get passed through once. In addition, most puzzles will only have a single solution, and while some authors choose to ignore either that constraint or the 'all cells get used' constrain, the majority of puzzles follow both.

Despite the name, Numberlink puzzles do not require the involvement of actual numbers. Instead, a puzzle may involve colored cells, symbols, or letters to indicate pairs that need connecting. As long as there are no duplicates, a traditional Numberlink can use anything to indicate matching cells.

When using numbers, puzzles also technically don't need to include unique pairs. One variation of a numberlink has numbers indicate the distance travelled by the path between pairs. This method allows for multiple pairs using the same number to occur, since their placement may be able to prevent ambiguity.

Examples

 * - A Build-Your-Own-Numberlink. Once arranged properly, the horizontal strips in this puzzle form a fully-solvable slitherlink using the numbers from 1 to 21.
 * - While the grid isn't provided in this one, there is one present. It just happens that it's represented by a dark maze of 104 cells. After some spoiler-y steps, solvers get the prompt to solve the grid as a Numberlink, but if they were to do that normally, it wouldn't have a unique solution. Instead, solvers need to treat it as a fixed-length Numberlink, where the content of each end of the path helps to indicate how long the path should be.
 * - A series of minipuzzles, the first of which is a Numberlink. However, it doesn't use numbers, instead choosing to connect colors. Once again, in order for it to be unique, it needs to be treated as a fixed-length. In this case, the length is provided by the names of the colors (which are all possible colors for a Nintendo Gamecube to be).