335 words on Uni
The kids are in our department again.
In an effort to popularise mathematics among school children, our department offers local schools to come by with interested pupils to get a little tour of some nice maths problems. This is about ‘grasping’ mathematics, in the literal sense of the word as most of the problems are little games with wooden parts and so on which the kids can touch, rearrange and so on to actually solve the problem. Even the little third graders who were here yesterday where surprisingly enthusiastic at this and did very well. Which is nice.
Problems include the classical Tower of Hanoi and those puzzles where you have a ring and some wooden blocks all joined up with a rope and have to find a way to ‘free’ the ring out of the structure. This looks impossible at first, but with a bit of tangling and untangling it can be done because, mathematically, things aren’t actually tangled but only look that way. That’s what we call knot theory or, more generally, topology.
I’ll present another topology problem here which is my favourite of the problems we present. It’s wrapped in a little story:
Your aunt gave you a painting. You don’t particularly like that painting. But when she comes to visit you’ll have to put it up to not disappoint her. She’s also a very stingy aunt and you know that when she sees a painting being fixed with two nails, she’ll pull one out to use it for something else. Wouldn’t it be nice if you could hang the painting in such a way using two nails that it falls down and breaks when your aunt pulls out one of them? That way you’d be rid of the painting and your aunt couldn’t complain.
Try and figure that one out and give a solution. It’s a fun problem. It’s much easier to get a piece of thread and actually do this. I might explain how to get a solution later on.
seems like a pretty simple problem to me… just make sure that the two nails are farther apart then the distance from the nails to the floor
That’s a fun idea, but not the one I was looking for.
The painting is supposed to really fall down, i.e. all none of the thread used to old it should fall down as well.
You could put the two nails inder the picture, holding it up..
Tom, that’d be cheating as well ;)
It’s perfectly possible to do this without using any tricks.
As a first step, try to hang the painting in a way that it always falls down when the right nail is removed.
Aaah, ok. You want a loop that isn’t actually round either of the nails. So you can solve the simplified problem by taking a loop off the top of the picture, put both threads over the left nail, and loop it over the right nail.
ah, ok. You want something where the thread isn’t actually over the nail it goes over the top one way, then over again the other way, but the other nail is in between, so removing either nail means that the remaining nail isn’t topologically involved any more..
I need to think more. My house really needs a whiteboard.
Oooh, that’s a lovely problem. Now I’m going to go annoy all my friends with it.
Hehe. that’ll be fun.
Drawing this – or simply trying it out makes the whole thing much easier. I’ll put up a post with the solution that contains some sketches and a hint towards the mathematics involved.
Instead of a whiteboard, may I recommend a blackboard?
(Hm, interestingly, whiteboard is a spelling mistake while blackboard isn’t. An oddity? Reasonable software?)
tom: try to hang the painting in a way that it always falls down when the right nail is removed
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