 # Sylvy’s Weekly Puzzle #3, Solution (part II)

Cross-reference to puzzle page on my website

In this post I want to carry on discussing the solution to puzzle #3. Recall that $\mathcal{S}$ denotes the world that Sonic explores in the Special Stage’. We are thinking of $\mathcal{S}$ as a topological space, upto topological equivalence.

We have already taken a look at the cases of the sphere and the torus, and we had shown that $\mathcal{S}$  cannot be a sphere and that $\mathcal{S}$ could be a torus. In fact we saw the map’ of the level: so it really IS a torus! But the question is: what can we say about $\mathcal{S}$ using only the topological information that sonic can gather?’

The final part of puzzle #3 asked us to see if $\mathcal{S}$ could be a Klein Bottle. A Klein bottle is the topological space obtained by taking a square and gluing together the opposite edges, as in the following diagram on the left. Note that the blue pair are glued together with a twist, while the red pair are glued straight’.

Well, the answer is yes’. Let $m,n\in\mathbb{N}$ be such that $m$ is even and $n$ is odd. Let $C$ be a  rectangle which is divided into an $m\times n$ chessboard, coloured as usual so that white squares and black squares alternate. I think of the bottom’ or horizontal’ edges as being $m$-squares long, and the side’ or vertical’ edge as being $n$-squares long. Below on the right is a picture for the case $m=5,n=6$.
Gluing the edges of this chessboard together in the way described above results in a Klein bottle which is tiled in the usual alternating black and white’ pattern that Sonic sees all around him. This shows that $\mathcal{S}$ could be a Klein bottle.