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

Cross-reference to puzzle page on my website

I’m really pleased at the interest the third puzzle has generated!¬†Or maybe I just haven’t been able to stop talking about it… ūüôā¬†Anyway, here is the first part of the solution:

The question asked us to investigate the world about which Sonic runs around in a certain special level. Let’s call that world \mathcal{S}. I want to describe¬†\mathcal{S} as a topological suface. If you’ve not come across this notion before, you might like to think of a few key examples:

  • the (real) plane \mathbb{R}^{2},
  • the open unit disk D^{2}=\{(x,y)\in\mathbb{R}^{2}\;|\;x^{2}+y^{2}<1\}, and
  • the sphere S^{2}=\{(x,y,z)\in\mathbb{R}^{3}\;|\;x^{2}+y^{2}+z^{2}=1\}.

Slightly more exotic examples:

  • the M√∂bius band,
  • the Torus,
  • (variations on the theme…) multi-handled tori, and
  • the Klein bottle.

Perhaps now isn’t the time for a complete exposition of topological surfaces (okay, I¬†may have got part-way¬†through drafting one…!), that may come later. For the present I just want to say a few words about the Euler characteristic¬†(which is denoted \chi) of a surface, and the formula that goes with it:

V-E+F=\chi\qquad(\star_{1}).

The first time I ever heard about this it was in the context of the Platonic solids, as follows.

  • Let’s begin by thinking of a cube. A cube has 8 corners (we’re going to call these vertices), 12 edges, and 6 faces. If V denotes the number of vertices, E the number of edges, and F the number of faces; then we have¬†V-E+F=8-12+6=2.
  • Next, let’s think of a tetrahedron. In this case there are 4 vertices, 6 edges, and 4 faces; thus we have¬†V-E+F=4-6+4=2.

(Can you see where this is going?)

  • Now let’s think of an¬†octahedron. Here there are 6 vertices, 12 edges, and 8 faces; we have¬†V-E+F=6-12+8=2.

You can check that the same formula

V-E+F=2\qquad(\star_{2})

holds for the other Platonic solids – but it doesn’t stop there. What about other solids, for example the Archimedean solid the¬†rhombicuboctahedron?

(I picked this one just because it has such a beautiful name.) The formula holds here too: we have V=24, E=48, F=26. Therefore (\star_{2}) holds for this solid:

V-E+F=24-48+26=2.

In fact:

The formula (\star_{2}) holds for all polyhedra that are topologically equivalent to the sphere \mathcal{S}^{2}.

Although I¬†won’t properly define `topological equivalence’ here, let me just say that – roughly speaking – two shapes A and B are topologically equivalent if one can be transformed into the other by continuous shrinking, stretching, bending, twisting, etc. (discontinuous transformations such as tearing or joining are not allowed). Each of the polyhedra discussed above is topologically equivalent to the sphere.

Let’s now think about the surface \mathcal{S} that sonic explores. It is given to us equipped¬†with¬†a¬†vertices/edges/faces structure so we can calculate V-E+F. On \mathcal{S},¬†each face has four edges and each vertex is the endpoint of four edges. We don’t seem to know how many faces there are, so let n be the number of faces. Then there are 2n edges and n vertices. [Why?] Thus $V-E+F=0$. This shows that \mathcal{S} is¬†not topologically equivalent to a sphere. This is a negative answer to the first part of the puzzle.

On the other hand, \mathcal{S} could be a torus, as shown by the following map for one of these levels:

[Map of a Special Stage in Sonic 3, copyright Sega (fair use of image for educational purposes)]

How does this show that \mathcal{S} could be a Torus? You can see quite easily that the programmers have joined the left edge to the right edge and joined the top edge to the bottom edge. Thus, if Sonic walks off the left edge of the map, he will just pass round to the right edge (without noticing!).  This shape is quite easily seen to be a Torus. This is a positive answer to the second part of the puzzle.

In the third part, I asked about whether \mathcal{S} could be a Klein bottle, and I’ll write a second post with the answer.

[Edit: perhaps it’s¬†unfair to use the map of the level that I found online to argue that \mathcal{S}¬†might be a Torus. In fact¬†the map (plus the description of how the edges are `obviously’ joined together) shows that \mathcal{S} definitely is¬†a Torus. If you didn’t have the map of the level then I think it’s reasonable to¬†simply¬†argue that a Torus can¬†admit a vertex/edges/faces structure as seen in the level (i.e. with `square’ faces joined in fours at vertices). We’ll address the question of whether \mathcal{S}¬†must be a Torus in the second part of this answer, when I’ll also discuss the Klein bottle.]

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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. ūüôā

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 http://anscombe.sdf.org/puzzle.html

Have fun!

Still mostly maths

Originally, I had intended this blog to be a tool I would use¬†to help teach an introductory course on Statistics, but that plan changed when I¬†switched to using UCLan’s own system (`Blackboard’) for managing course materials. So this blog has a new purpose: it will be a random, unplanned, haphazard collection of bits and bobs that pique my interest…. but it will always¬†be¬†`mostly maths’.

 

Hello! A new semester at UCLan

Hello! I have decided to write a blog to accompany my teaching this semester at UCLan. I will be teaching MA1861 Introductory Statistics, and I hope that this blog will provide a forum for additional discussion and explanation of topics contained in the course and some beyond it…

This blog will also host discussions of other topics (mostly mathsy) which take my fancy.

The next post, to be written shortly, will give a provisional course outline.

Edit: a slight change of purpose. There will be a blog on the `blackboard space’ for MA1861 (at http://portal.uclan.ac.uk) for official course content. Instead, this blog will contain various mathematical-bits and sundry logical-bobs; some related to teaching and others not. But nothing that is not written on the blackboard space for MA1861 will be required reading for that module. Inevitably there will be some cross-posting.