Sylvy’s weekly puzzle #3

Cross-reference to puzzle page on my website

My students already know that I am setting `weekly’ fun maths puzzles – although they’re not quite weekly! The first two can be found here, and I may well elaborate on them in later posts. For now, I want to discuss puzzle number 3. Remember: this is aimed at undergraduates in the first term of their first year, so please: no spoilers from more knowledgeable folks. 🙂

Sylvy’s Weekly Puzzle #3

This one is a little more `open ended’ than the previous puzzles. The winner will be the best attempt at an `investigative’ solution.

I tried to describe one of my favourite levels on the computer game Sonic to you (in fact I think it was on Sonic 3 – my bad), here is a screenshot:

[Special stage on Sonic 3 Copyright Sega (fair use of image for educational purposes)]

The game is played on a certain kind of `grid’. It’s an infinite two-dimensional grid, like a chessboard but infinite in all directions. No matter where Sonic goes, his world looks like this. My question is:

What can you say about the topology of his world?

Of course, that’s really an unfair question, because it should be asked to students that have studied a bit of topology or graph theory. Let me rephrase it more simply: could the world Sonic inhabits in the bonus level be:

(i) a sphere?

(ii) a torus?

or (harder)

(iii) a Klein Bottle?

Key things to investigate here are: Euler’s formula \chi=V-E+F and the classification of closed surfaces.

Deadline: 10am Thursday 22nd October (either by email or hand-written solutions in the folder on the outside of my office door)

Prize: something edible (several of the prizes from Puzzle #4 onwards will include a book of Mathematical Puzzles!)

Don’t forget: Class at 10am THIS THURSDAY in my office to go over the previous puzzle.

The webpage for these puzzles is

Have fun!