The Geek Forum
Main Forums => Anarchy => Topic started by: ivan on February 02, 2011, 05:34:55 PM
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If you haven't cracked Jaepheth's puzzle yet, and still plan on trying, then don't read beyond this post.
Ok, then.
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Ok, since this is a pretty difficult problem I'll give you all a Big Hint (http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality)
Hint, shmint. What are you trying to pull here? The dudes pay the same price every week, so over time they get the same deal. Anyone can see that!
Here, I will prove it:
Week 1: Price $5 per kilo. Adam buys 1 kilo for $5, Bob buys .2 kilos for $1
Week 2: Price $4 per kilo. Adam buys 1 kilo for $4, Bob buys .25 kilos for $1.
Week 3: Price $8 per kilo. Adam buys 1 kilo for $8, Bob buys .125 kilos for $1.
Week 4: Price $2 per kilo. Adam buys 1 kilo for $2, Bob buys .5 kilos for $1.
Total: Adam bought 4 kilos for $19. Bob bought 1.075 kilos for $4.
In aggregate, Adam paid $4.75 per kilo, and Bob paid exactly the same, $3.72 per kilo.
Oh shit.
Common sense didn't work out so well.
Another thing common sense tells me is that one method of buying should be advantageous when prices are rising, and a disadvantageous when prices are falling. But that also appears to be false.
So, about that Schwarz Inequality... how does it work?
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Strictly speaking, the Cauchy-Schwarz inequality isn't absolutely necessary to solve the problem. Perhaps I just felt like being mean.
But what if there were two n dimensional vectors, the lengths of which described Adam and Bob's respective averages?
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Now you're just trying to act like a stupid-face smarty pants. :slap
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Now you're just trying to act like a stupid-face smarty pants. :slap
^
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I gotta say I'm impressed, however.
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Ok, Here's another tidbit
When I said I was being mean. That was a clue (http://en.wikipedia.org/wiki/Mean)
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Ok, Here's another tidbit
When I said I was being mean. That was a [http://en.wikipedia.org/wiki/Mean]clue[/url]
What is this I don't even
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Ok, THIS (http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means) is the crux of the proof. Now you just have to algebra the numbers to show who is paying what mean.
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Hey... I knew this was familiar. There was a Jensen's Inequality problem that had to do with something like this, years and years ago when I was still in school.