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  • sylvyanscombe 6:31 pm on November 10, 2015 Permalink | Reply
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    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).

    (More …)

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  • sylvyanscombe 10:00 am on November 10, 2015 Permalink | Reply
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    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’.

    Gluing the edges of a square to form a Klein Bottle

    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.

    even-by-odd chessboard

    even-by-odd chessboard

    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.

     
  • sylvyanscombe 9:16 pm on November 4, 2015 Permalink | Reply
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    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!!

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

    Have fun!

     
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