Solution to Slitheɹlᴉuʞs

As clued by the title and flavortext, these Slitherlink variants take place on a Möbius strip. Each strip should be joined with a half twist such that red and blue arrows line up. Notice that when solving on a Möbius strip, the Slitherlink loop may or may not be contractible. So in contrast with regular Slitherlinks where the number of horizontal segments in each vertical must always be even, the number of horizontal segments here may either always be even or always be odd. Three Slitherlink variants are given and their solutions are shown below.

While solving the Slitherlinks, we might notice that there are additional diagonal red and blue lines that have not yet been used. When the Slitherlinks are printed and assembled as Möbius strips, each red line is on the opposite side of a blue lines. The type of line dash pattern on the red lines indicate they are mountain folds and the line dash patterns on the blue lines indicate they are valley folds. Each Möbius strip can be folded along the 3 folds and flattened to form a shape similar to the universal recycling symbol. When the Möbius strips are folded and flattened, the solution the Slitherlink traces out a character on each flattened face. Below is a color coded Möbius strip and the flattened faces with the Slitherlink solutions traced.

This string of characters can be parsed as per the enumeration given by the blanks at the bottom of the puzzle page. Doing so gives us MU FUNC OF LENGTH MOD 3. Here, MU FUNC refers to the thematic Möbius function which is often denoted with the Greek letter mu. The Möbius function takes as input positive integers and returns either 0, 1, or -1. Applying the Möbius function to the length of loop in each Slitherlinks solution and taking the value modulo 3 gives us the following values.

Loop Length	Möbius function value modulo 3
86	1
118	1
138	2
128	0
122	1
154	2
114	2
124	0
130	2

Each Slitherlink puzzle extracts to either 0, 1, or 2 and there are 3 Slitherlink puzzles per Slitherlink variant. This suggests extraction via ternary. Converting the numbers 112, 012, 202 in ternary produces the answer NET.