## Puzzle corner: Sylvy’s puzzle #5

Cross-reference to puzzle page on my website

Here is my next puzzle! Sorry that it’s been greatly delayed.

I mentioned previously the problem of constructing the midpoint of a given circle, using only ruler and compass. One student solved it nearly immediately (congratulations!), so here I offer a more general problem.

#### Sylvy’s puzzle #5

Given a parabola in the plane, construct its vertex using straightedge-and-compass construction (sometimes called ruler-and-compass’ construction).

## 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.

Note that we’ve only taken into account topological information. In later posts I want to tackle the geometric’ question.

## Sylvy’s weekly puzzle #4a

Cross-reference to puzzle page on my website

It seems that my first-year students (who form a large part of the target audience for my puzzles) are unfamiliar with the countable’ and uncountable’ terminology that I used in puzzle #4. …oops!

Instead of introducing it now, I’ll give a completely different puzzle which was originally shown to me by a friend (more on this later).

Sylvy’s weekly puzzle #4a

Show by means of a `geometrical drawing’ that

$\arctan(1)+\arctan(2)+\arctan(3)=\pi$.

deadline: 11am Monday 9th of November

how to hand-in: send typed solution by email or put hand-written solution into the folder on my office door

prize: I’m very pleased to announce that the winner of this puzzle will get a book of mathematical puzzles!!

Have fun!