# Monthly Topic Challenge #10: Möbius Strips, Klein Bottles, and other unusual topological surfaces

This is the tenth installment of the Monthly Topic Challenges with topics suggested and voted on here. This month's topic is "Möbius Strips, Klein Bottles, and other unusual topological surfaces" (suggested by melfnt) and will span from the 1st of May to the 31st of May. During this period, we will compile the list of relevant questions and post it as an answer to this question.

In the meantime, please go and propose and vote on future challenges!

Everyone have fun, and happy puzzling!

Link to other Monthly Topic Challenges.

NOTE
The suggestion is copied to this post for posterity.

## Möbius Strips, Klein Bottles, and other unusual topological surfaces

Imagine an elastic square as above. If you ignore the red edges and glue the blue edges so that they have the same orientation, you get a Möbius Strip. If you join both pairs of edges, you get a Klein Bottle. This is not the only interesting surface; you can have 12 different topologies in total by joining two pairs of edges of a square, which include:

(The last one is a plain torus.) The universal rule is that, if you're on the surface and you walk through an edge, you enter back to the grid through the matching edge with same orientation.

For example, consider this 4x4 grid:

A1 A2 A3 A4B1 B2 B3 B4C1 C2 C3 C4D1 D2 D3 D4

Assume the Klein bottle topology at the top. If you're at B1 and walk left, you end up at B4 (by going through the red parallel edge). If you're at A2 and walk upwards, you end up at D3 (by going through the blue twisted edge).

For the first one on the second image set (the one where the matching edges are adjacent to each other), if you walk upwards from A1, you enter back at A1 but facing right!

The challenge is to create a puzzle that involves an unusual topological surface (which excludes plain wrap-around mechanic a.k.a. cylinder and torus). Such a puzzle may involve tiling, graph, crossword, or a grid-deduction genre, among others.

Some good examples:

You can search for puzzles containing torus, toroidal, or wrap around for some ideas.

There is even a programming language called Klein that can run the same code on different topologies!

copied from Fortnightly Topic Challenges Threrun: Topic Suggestions originally posted by Bubbler