30-07-2004 08:55									#91
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quote: Originally posted by Kuciwalker Maybe we should be more like Aristotle's lyceum: just math and philosophy

http://www.cals.ncsu.edu/agexed/aee501/aristotle.html

'All students were assigned tasks involving historical or scientific research. Aristotle based many of his scientific propositions upon this research.'


http://www.edst.purdue.edu/georgeof...0/Aristotle.htm

'Although study at the Lyceum seems to have concentrated on rhetoric and logic, a shift shortly was made to the natural sciences, resulting in a curriculum distinctly different from Plato’s Academy.'

Including but not limited to, zoology, marine biology, cosmology, taxonomy, Aristotelean (non-mathematical) logic....

http://people.bu.edu/wwildman/Weird...hil_theme03.htm

'Aristotle’s Lyceum

Founded in 335, close to Athens, when Aristotle was 49
Curriculum emphasized natural science; contrast with Academy’s stress on mathematics, and speculative and political philosophy. '

	30-07-2004 08:57									#92
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20101?

Thanks, jag. You may well be better than me at this sort of thing - I know almost nothing about combination-stuff. This is a rough guess at a patter that I think exists

	30-07-2004 08:59									#93
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quote: Originally posted by Jaguar I have never solved a problem like this one

What the **** did you just say to me?

Goddamnit, neither had I until I saw it about 2 days ago.

Mathematics is not about applying known methods. Exercise your goddamn creativity. If you can't do that then you will never progress beyond mere rote ability.

And the people who write those math competitions are creative enough in coming up with problems that you will never be able to excell via knowledge of methods...

	30-07-2004 09:00									#94
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quote: Originally posted by molly bloom http://www.cals.ncsu.edu/agexed/aee501/aristotle.html 'All students were assigned tasks involving historical or scientific research. Aristotle based many of his scientific propositions upon this research.' http://www.edst.purdue.edu/georgeof...0/Aristotle.htm 'Although study at the Lyceum seems to have concentrated on rhetoric and logic, a shift shortly was made to the natural sciences, resulting in a curriculum distinctly different from Plato’s Academy.' Including but not limited to, zoology, marine biology, cosmology, taxonomy, Aristotelean (non-mathematical) logic.... http://people.bu.edu/wwildman/Weird...hil_theme03.htm 'Aristotle’s Lyceum Founded in 335, close to Athens, when Aristotle was 49 Curriculum emphasized natural science; contrast with Academy’s stress on mathematics, and speculative and political philosophy. '

Sorry, I got the two mixed up. I'm not an enyclopedia

	30-07-2004 09:00									#95
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quote: Originally posted by Kuciwalker 20101? Thanks, jag. You may well be better than me at this sort of thing - I know almost nothing about combination-stuff. This is a rough guess at a patter that I think exists

Closer. You're off by just under a factor of two.

Where does the number 20101 come from (i.e. what numbers did you multiply together to get that)?

If you tell me I may be able to reverse engineer your mistake

	30-07-2004 09:01									#96
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quote: Originally posted by KrazyHorse Hint: the prime factorisation of 2004^100 is: 2^200 * 3^100 * 167^100

It's easier if you just think of it as 2x2x3x5 (^100) though - the answer is the same whether you use 5 or 167, and 5 is easier to write

	30-07-2004 09:02									#97
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quote: Originally posted by Kuciwalker Sorry, I got the two mixed up. I'm not an enyclopedia

Neither am I.

	30-07-2004 09:02									#98
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quote: Originally posted by KrazyHorse Closer. You're off by just under a factor of two. Where does the number 20101 come from (i.e. what numbers did you multiply together to get that)? If you tell me I may be able to reverse engineer your mistake

200+199+198...+1 (and then tacked on an extra 1 for (3^100)*(5^100))

	30-07-2004 09:02									#99
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quote: Originally posted by molly bloom Neither am I.

So you just copy and paste from, say, Grolier?

	30-07-2004 09:03									#100
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Obviously.

The really important part is that one of them is to the two hundred whereas the others are to the one hundred....

	30-07-2004 09:05									#101
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quote: Originally posted by Kuciwalker 200+199+198...+1 (and then tacked on an extra 1 for (3^100)*(5^100))

Look more carefully at the pattern that you've built

Some of those numbers are dividing each other...

In fact, if you've built it according to how I think you did, an awful lot of them are...

	30-07-2004 09:12									#102
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The question can be though of as follows:

assign A a number from 0 to 200, B a number from 0 to 100 and C the same

Do it many times in such a way that between any two assignments neither is larger (or equal) in all three categories

Build starting with C = 0

Obviously every other C will be larger, so we need to make either A or B bigger than it will be with any assignement. The way to do this is to start with B at 0. Now it is best to take A = 200 Make every possible combination such that A + B = 200 and you'll be safe so far

There are 101 of these (200,0 199,1 etc)

Now raise C to 1 and find every possible combination st A+B = 199 (101 of these)

Keep going until C = 100

So the answer is 101*101 = 10201

In other words, you found every possible combination such that A + B + C = 200

As an additional exercise, extend this thought to an arbitrary prime factorisation...

	30-07-2004 09:14									#103
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The proof that this is the largest possible applicable subset is substantially more involved. It took me about an hour to see it. My proof is rather involved. There might be a shorter one, but I haven't found it yet...

	30-07-2004 09:14									#104
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spoiler(highlight to read): 10202?

	30-07-2004 09:14									#105
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quote: Originally posted by Kuciwalker So you just copy and paste from, say, Grolier?

No, I strive for accuracy.

	30-07-2004 09:15									#106
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quote: Originally posted by KrazyHorse Look more carefully at the pattern that you've built Some of those numbers are dividing each other... In fact, if you've built it according to how I think you did, an awful lot of them are...

I forgot that I had a limited number of 3's and 5's to play with.

	30-07-2004 09:15									#107
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See above. Almost....

	30-07-2004 09:16									#108
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Didn't you forget (3*5)^100?

EDIT: I originally guessed it would be the 101st square, but I thought that was too easy and tried to work it out on paper some more

	30-07-2004 09:19									#109
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No, that's included in it...

In fact, it's included as the last possible combination

	30-07-2004 09:20									#110
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BTW, this was not a multiple choice test. Papers must have justification for their answers...

	30-07-2004 09:20									#111
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Oh, I see.

	30-07-2004 09:22									#112
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Not bad, by the way Kuci

Pattern-building is the most important analytical tool a mathematician possesses.

No proof is built as a computer would build it. Every theorem exists because somebody noticed a pattern in examples they were creating and tried to prove that the pattern was a general principle. Intuition first, logic second.

	30-07-2004 09:23									#113
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There's a reason we haven't built electronic mathematicians yet.

We just let them help us on stuff like the 4 colour theorem...

	30-07-2004 09:23									#114
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I once spent about 2 hours staring at a list I had put together of the first 100 primes, powers of 2, a bunch of other numbers, etc. looking for a pattern. I nearly drove myself nuts

	30-07-2004 09:24									#115
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quote: Originally posted by KrazyHorse There's a reason we haven't built electronic mathematicians yet. We just let them help us on stuff like the 4 colour theorem...

We'll see after I study AI next year

	30-07-2004 09:26									#116
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Good luck. Mathematics (and physics) have an art to them, as well as a science...

We can build a computer than can beat the best chess players in the world.

I think it will be a while before we build a computer that can replace a great mathematician.

	30-07-2004 09:26									#117
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I'd rather have a computer that was good at playing civ

	30-07-2004 09:29									#118
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quote: Originally posted by Kuciwalker I once spent about 2 hours staring at a list I had put together of the first 100 primes, powers of 2, a bunch of other numbers, etc. looking for a pattern. I nearly drove myself nuts

I drove myself insane trying to prove that given a point and a line, there's only one parallel of the line that goes through the point.

It's unprovable.

	30-07-2004 09:30									#119
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Hyperbolic geometry

	30-07-2004 09:30									#120
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You and I both know that if they put their minds to it they could build one that would beat all of us.

The problem with civ (as compared with chess) is that the game is hopelessly complex. I think that this complexity, however, degrades the additional benefit of the unique best move. I think a computer's micromanagement skills would destroy the best human player's setup if programmed any way better than half-assed (i.e. what we have so far)