Science and Technology Quiz 3

GoodGame · Apr 28, 2009

ParadigmShifter said:
OK. John Nash was one of the inventors of a board game which can never end in a draw.

What game is it, and proving that the n-dimensional variant can't end in a draw either also proves a famous theorem, what theorem is that? And what has it got to do with dropping a map of the Earth on the floor?

On the last point, I'd guess the theorem has a practical importance to map projections.

Hmm. Checkers and Chess can stalemate. I think the game is that one you play on diner menus, with the playing field a grid of dots, and each player adds one line connecting to adjacent dots at a time. IIRC, the object is to make the most complete boxes, with each box giving one score.

EDIT: I googled his name on Wikipedia, and they only list is works. The topics got me thinking some more. Would his game be Chinese Checkers?

ParadigmShifter · Apr 28, 2009

Nope it aint that game.

EDIT: And the theorem isn't to do with map projections.

ParadigmShifter · Apr 28, 2009

GoodGame said:
EDIT: I googled his name on Wikipedia, and they only list is works. The topics got me thinking some more. Would his game be Chinese Checkers?

Nope, that was before Nash was born (he's still alive in fact).

EDIT: I deliberately left out the name of the game since wiki'ing tells you all the answers

GoodGame · Apr 28, 2009

ParadigmShifter said:
Nope, that was before Nash was born (he's still alive in fact).

EDIT: I deliberately left out the name of the game since wiki'ing tells you all the answers

That's ok, for some reason I wiki'd John Casey. LOL.

sanabas · Apr 28, 2009

ParadigmShifter said:
OK. John Nash was one of the inventors of a board game which can never end in a draw.

What game is it, and proving that the n-dimensional variant can't end in a draw either also proves a famous theorem, what theorem is that? And what has it got to do with dropping a map of the Earth on the floor?

A Danish bloke invented it first. It's mostly called Hex, and the theorem is some sort of fixed point theorem that I can't actually remember.

Truronian · Apr 28, 2009

Played-by-Russell-Crowe-in-A-Beautiful-Mind John Nash?

ParadigmShifter · Apr 28, 2009

Yeah that's him.

ParadigmShifter · Apr 28, 2009

sanabas said:
A Danish bloke invented it first. It's mostly called Hex, and the theorem is some sort of fixed point theorem that I can't actually remember.

I'm only going to accept all the answers in a single post.

sanabas · Apr 28, 2009

Truronian said:
Played-by-Russell-Crowe-in-A-Beautiful-Mind John Nash?

Yes, and in the book too. Though I think the game was called Nash in the movie/book. They filmed half that movie inside my head.

I'm only going to accept all the answers in a single post.

I could look up the actual details of the theorem, but that would be cheating. As for what it has to do with dropping a map of the earth on the floor, that would be that some point on the map will be in exactly the same spot as the place it represents. Which doesn't actually help me with the details of the theorem.

Also trying to remember the one that proves that if you have a continuous variable, say temperature, or height above/below sea level, and a closed loop of points, such as a great circle around the earth, there will be two points opposite each other on the circle that have exactly the same value for whatever variable you're looking at. Will have to look that one up and/or do the proof again.

ParadigmShifter · Apr 28, 2009

OK, I'll give it to you sanabas.

It's called the Brouwer fixed point theorem and as you pointed out applies in more dimensions than just 2.

Proving that Hex can't be a tie proves the 2D case, the higher dimensional variants prove it in general.

The reason a map dropped on the floor (if your house is located somewhere on the map) will have a point in the correct place is because the map is smaller than the world.

If the map is larger than 1:1 scale that isn't necessarily true.

EDIT: Your question about fixed points on a sphere is known as the Hairy Ball Theorem

http://en.wikipedia.org/wiki/Hairy_ball_theorem

sanabas · Apr 28, 2009

Your edit was going to be my next question, in order to save me searching. Which may mean you're developing psychic powers, and may also mean you're up again.

ParadigmShifter · Apr 28, 2009

Nah I can't answer the question after mine, so you have to think of another

sanabas · Apr 28, 2009

ParadigmShifter said:
If the map is larger than 1:1 scale that isn't necessarily true.

It should be. Because you're essentially dropping your floor onto the map, so as long as all the bits of map you're landing on represent your floor, then the theorem works exactly the same way.

For it to not necessarily be true, you need to be dropping part of it somewhere not covered in the map.

*ninjaedit* pffft. Will see what question I can dream up then. */edit*

ParadigmShifter · Apr 28, 2009

sanabas said:
It should be. Because you're essentially dropping your floor onto the map, so as long as all the bits of map you're landing on represent your floor, then the theorem works exactly the same way.

For it to not necessarily be true, you need to be dropping part of it somewhere not covered in the map.

Hmm I'll have to think about this. I thought you could have a translated point instead of a fixed point in that case.

It's also called the Contraction Mapping Theorem because the scale of the transform must be < 1.

sanabas · Apr 28, 2009

OK, really really easy one about my favourite number. For the series 3, 7, 10, 17, 27, 44, 71, ..., what number does T(n+1)/T(n) converge to (exact value please)? What else is this number called, and what's so interesting about it?

sanabas · Apr 28, 2009

ParadigmShifter said:
Hmm I'll have to think about this. I thought you could have a translated point instead of a fixed point in that case.

It's also called the Contraction Mapping Theorem because the scale of the transform must be < 1.

I'll have to read up too. But if the transform from real world to map is scale > 1, then the transform from map to real world is scale < 1. If you imagine the map as the real thing, and the bit of world the map is landing on as the smaller scale model, then provided all the bits of the map landed on are on the smaller scale model, you'll still have a fixed point. Same as when you drop the smaller scale map, provided everywhere the model lands on is on the model, you'll have a fixed point.

dutchfire · Apr 28, 2009

The sequence is: T(n)=T(n-1)+T(n-2)
It converges to 1,618034, which is, if I remember correctly, Phi, <insert DaVinci code crap about the importance of phi in nature/art/everything here>.

I also remember that it has got something to do with sqrt(5), but I don't know how exactly.

sanabas · Apr 28, 2009

Exact value please, and DaVinci code crap can be safely left out, reading that book was a complete waste of 3 hours, it's tripe.

dutchfire · Apr 28, 2009

Some fooling around with Excel suggests (1+sqrt(5))/2

sanabas · Apr 28, 2009

Yeah, it is. Though proofs are much shinier than trial and error via excel.

You are up, and everyone else should type 'golden ratio' into google, and discover lots of interesting stuff.

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