Zig-Zag 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: (i) the endpoints of the $i$-th path are the two cells containing number $i$, and (ii) each cell of the grid belongs to exactly one path.

A popular smartphone game based on Zig-Zag Numberlink is Flow Free.

The problem of deciding whether an instance of Zig-Zag Numberlink admits a solution is NP-Complete [1].

The variant in which each path is required to use the minimum number of turns when considering the other paths as fixed is known as Numberlink.

[1] A. Adcock, E. D. Demaine, M. L. Demaine, M. P. O’Brien, F. Reidl, F. S. Villaamil, B. D. Sullivan, “Zig-Zag Numberlink is NP-Complete”, Journal of Information Processing, 2015.

@misc{cog:zig-zag-numberlink, author = "{CoG contributors}", title = "{Zig-Zag Numberlink (Flow Free) --- Complexity of Games}", year = "2020", url = "https://www.isnphard.com/i/zig-zag-numberlink/", note = "[Online; accessed 2020-01-03]" }