Numberlink is a single player game played on a rectangular grid. Initially, some of the cells of the grid are empty while others contain a number from $1$ to $k$, each of which appears exactly twice.
Each grid cells is adjacent to the cells to its left, right, top, and bottom, if any. The goal is to find a collection of $k$ simple paths (sequences of adjacent cells) such that:
The problem of deciding whether an instance of Numberlink admits a solution is NP-Complete .
The variant in which condition 3 is removed is known as Zig-Zag Numberlink.
 K. Kotsuma, Y. Takenaga, “NP-Completeness and Enumeration of Number Link Puzzle”, IEICE Technical Report, 2010.