maths problem for DS

[chezlaw]

[ QUOTE ]
Could you post some problems for skalansky to break down into simple logic?

Skalansky flexes his math logic alot so i think it would be interesting to see if he approaches things as uniquely as he makes it sounds like he does.

[/ QUOTE ]
here's a very simple maths problem, so simple a kid can solve it. (so simple there's a danger DS may solve it with recourse to nothing but simple logic). Hopefully it will illustrate the point.

What's the sum of the first n natural numbers? i.e

1+2+3+....+n =

[soon2bepro]

This is what I came up with: (in white)

<font color="white">
1 | *1
2 | *1.5
3 | *2
4 | *2.5
5 | *3
6 | *3.5
7 | *4
8 | *4.5
9 | *5
10 | *5.5
11 | *6

and goes on forever...

That means that n plus all the numbers before it = n [(n/2) + 0.5]

So for example the sum of

1+2+3+4+5+6+7+8+9+10 = 10 * [(10/2) + 0.5)

= 10 * 5.5
= 55 </font>

[chezlaw]

[ QUOTE ]
This is what I came up with: (in white)

<font color="white">
1 | *1
2 | *1.5
3 | *2
4 | *2.5
5 | *3
6 | *3.5
7 | *4
8 | *4.5
9 | *5
10 | *5.5
11 | *6

and goes on forever...

That means that n plus all the numbers before it = n [(n/2) + 0.5]

So for example the sum of

1+2+3+4+5+6+7+8+9+10 = 10 * [(10/2) + 0.5)

= 10 * 5.5
= 55 </font>

[/ QUOTE ]
but was that by simple logic? Yes its simple but thats because its a simple problem.

Break it down into the steps you made and the reasoning behind them.

chez

[David Sklansky]

First off I think you misunderstood my original comment about math and logic. I didn't mean that advanced problems can be easily solved without recourse to advanced math. I meant that they could in theory be solved that way. After all advanced math is just a logical progression from less advanced math. (Which is why it is ridiculous to postulate someone who is extremely adept at everything through calculus could have more trouble with higher math than someone who wasn't.)

Meanwhile unless you have some trick up your sleeve your problem is a perfect example of my point. Add them up two at a time from the outside inwards. n/2 pairs of n+1 totals.

[nightwood]

It's idle to debate whether a step is "simple" or not; for some it will be for some not, depending on various factors.

Once a problem is solved and the proof is tested it's quite easy to say "Aw, yes, all but simple steps." - now, getting to this solution is a completely different thing.

Let's just take a slightly more difficult problem - Euclids Proof (http://en.wikipedia.org/wiki/Prime_numbers#There_are_infinitely_many_prime_numb ers) of the infiniteness of prime numbers. It's very easy to grasp .. but I'm quite interested if anyone's willing to argue that it's easy to come about it.

[chezlaw]

[ QUOTE ]
Meanwhile unless you have some trick up your sleeve your problem is a perfect example of my point. Add them up two at a time from the outside inwards. n/2 pairs of n+1 totals.

[/ QUOTE ]
That requires an idea. Once you've had the idea the rest is simple logic. Many people would never have that idea but can grasp it if its pointed out because integers are simple and its a simple idea.

To the point, the idea and understanding are not reducible to simple logic and its the inability to grasp the concepts and ideas that prevents people from doing advanced maths.

All of us are limitation in this respect and its why although anyone of moderate intelligence can do maths methods (differential equations etc) most struggle with advanced abstract algebra or number theory - these require very high intelligence to graps the concepts and ideas.

chez

[soon2bepro]

[ QUOTE ]
Break it down into the steps you made and the reasoning behind them.

[/ QUOTE ]

Ok, first I made a list of n and the result up to n11, calculating each result independently.

It looked like this:

1 | 1
2 | 3
3 | 6
4 | 10
5 | 15
6 | 21
7 | 28
8 | 36
9 | 45
10 | 55
11 | 66

Then I tried to spot some pattern in them. Some formulae that would work for all of them, depending on n of course. I started finding patterns until about 5 minutes after starting I got the whole deal and broke it down into the math.

At first I noticed that the odd numbers's result was a multiple of n. for 1, it was 1. for the second odd number, 3, it was 2. For 5, it was 3, and so on. Then I noticed that 1.5, 2.5 etc, worked for the even ones. It was all pretty simple from there.

[bigpooch]

Technically, if n is odd, you won't have "n/2 pairs".

A very simple solution, one that most children understand,
is based on the commutativity of addition and is really the
same idea:

1+2+3+...+n = S
n+(n-1)+...+1 = S

Adding the two equations above,

(n+1)+...+(n+1) = 2S

or n(n+1) = 2S
or S = n(n+1)/2.


IDEAS
=====

An idea can be very simple (such as above) or complex (such
as Kummer's idea of numbers of the form a+bi in attempt to
solve FLT). Even when an idea fails to solve a problem, it
can create a body of theory. What differentiates great
problem solvers from ordinary ones is not only perseverance,
but finding the "key ideas". Often, solving problems
require many ideas; that is why mathematicians often need
to have exposure to analysis and abstract algebra as well as
their area(s) of research.

Also, some ideas for proofs are very beautiful, but often
require some background. For example, I am still enamored
with Furstenberg's proof of the infinitude of primes (based
on point-set topology), but obviously it is not as easily
accessible as the common proof attributed to Euclid.

[Enrique]

Furstenberg's proof is beautiful. Here's the link (http://en.wikipedia.org/wiki/Furstenberg's_proof_of_the_infinitude_of_primes)

[Enrique]

[ QUOTE ]
First off I think you misunderstood my original comment about math and logic. I didn't mean that advanced problems can be easily solved without recourse to advanced math. I meant that they could in theory be solved that way. After all advanced math is just a logical progression from less advanced math. (Which is why it is ridiculous to postulate someone who is extremely adept at everything through calculus could have more trouble with higher math than someone who wasn't.)


[/ QUOTE ]

I wonder if anyone doubts that with time and work, one can understand higher mathematics. The problem is that it is a lot of work and it needs a lot of time.

I am very adept at everything through calculus, and very adept at higher math, but there's a point where going to much higher math needs a serious time commitment. To get depth, you need a lot of time and work, and there are so many branches in mathematics, that one can't really go into all of them.

Here's an anecdote from Von Neumann:
[ QUOTE ]
"It is often said that modern mathematics is so vast that no one can know more than a tiny fraction of it. Someone once asked von Neumann how much of mathematics he himself knew. Von Neumann went into one of his characteristic thinking trances. After a moment he had an answer. 'Twenty-eight percent.'"

[/ QUOTE ]

And that was one of the most versatile mathematicians and mathematics wasn't as broad as now.

(I probably went way off topic, but I don't understand what point you are trying to make).

[furyshade]

there is a simple way of doing this, its called a series. any series of all whole numbers from 1 to n can be represented as (n+1)n/2. simply you add the first and last number of a set of all whole numbers, and multiply by half the total number of integers in the set.

think about it like this, you have 1+2+3+4+5+6+7+8+9+10. you have 5 sets of number, 1+10, 2+9, 3+8, 4+7, 6+5, each equaling 11, or n+1. since there are half as many sets of numbers as there are number, you multiply n+1 by the number of sets, which is n/2.

[nightwood]

[ QUOTE ]
Furstenberg's proof is beautiful. Here's the link (http://en.wikipedia.org/wiki/Furstenberg's_proof_of_the_infinitude_of_primes)

[/ QUOTE ]

Very much OT but - it definitely is.

[Phil153]

[ QUOTE ]
First off I think you misunderstood my original comment about math and logic. I didn't mean that advanced problems can be easily solved without recourse to advanced math. I meant that they could in theory be solved that way. After all advanced math is just a logical progression from less advanced math. (Which is why it is ridiculous to postulate someone who is extremely adept at everything through calculus could have more trouble with higher math than someone who wasn't.)

Meanwhile unless you have some trick up your sleeve your problem is a perfect example of my point. Add them up two at a time from the outside inwards. n/2 pairs of n+1 totals.

[/ QUOTE ]
David,

The thing about math is that it's like a integrated circuit or branching tree, only many times more complex and interlinked:

http://img231.imageshack.us/img231/1874/20041018sz2.jpg

Each node is an area of mathematics, or a distinct mathematical concept. But the thing is, the human brain can only see this much at a time:

http://img217.imageshack.us/img217/8434/ftomsh5.gif

And can only learn a portion of the circuit in their lifetime.

When you're trying to find complex interrelationships, or prove the closed nature of link that branches in multiple directions or extends to infinity, or juggle multiple inputs to produce a useful output, it becomes much more than a simple game. There is actual genius and significant creativity required to achieve such things because the topic is so massively complex and abstract that you're operating on a different level of thought. Simple iterative logic doesn't get you anywhere past a certain point.

[David Sklansky]

[ QUOTE ]
[ QUOTE ]
Meanwhile unless you have some trick up your sleeve your problem is a perfect example of my point. Add them up two at a time from the outside inwards. n/2 pairs of n+1 totals.

[/ QUOTE ]
That requires an idea. Once you've had the idea the rest is simple logic. Many people would never have that idea but can grasp it if its pointed out because integers are simple and its a simple idea.

To the point, the idea and understanding are not reducible to simple logic and its the inability to grasp the concepts and ideas that prevents people from doing advanced maths.

All of us are limitation in this respect and its why although anyone of moderate intelligence can do maths methods (differential equations etc) most struggle with advanced abstract algebra or number theory - these require very high intelligence to graps the concepts and ideas.

chez

[/ QUOTE ]

As usual we have a communication problem. Because I agree with everything above. Which is why David Steele and myself called madnak's Fred and Ginger impossibilities. Fred wouldn't be able to perfectly answer all elemetary problems unless he had very high intelligence. And if he did he would have little trouble with tougher stuff.

(Even though most people like to pooh pooh the SAT and GRE math aptitude tests, I would guess that 80% of those who could get a perfect score on those tests at age 16 could go on to get a Phd in math if a gun was held to their heads while only perhaps 20% of those scoring 700 at age 18 could do that Phd. 700 still being in the top 3% or so. If I'm right it proves that there is an very high correlation between ability at simple math and higher math.)

[blah_blah]

[ QUOTE ]

Even though most people like to pooh pooh the SAT and GRE math aptitude tests, I would guess that 80% of those who could get a perfect score on those tests at age 16 could go on to get a Phd in math if a gun was held to their heads

[/ QUOTE ]

I have perfect scores on the aforementioned tests, a couple of top 200s on the putnam, a 98th percentile score on the GRE math subject test, am a math graduate student, and think that you are very, very wrong with this assertion.

I think that most people who could do this could probably get a degree in math from an average school and possibly a masters from a mediocre school, but a Ph.D is too ambitious.

[David Sklansky]

[ QUOTE ]
[ QUOTE ]

Even though most people like to pooh pooh the SAT and GRE math aptitude tests, I would guess that 80% of those who could get a perfect score on those tests at age 16 could go on to get a Phd in math if a gun was held to their heads

[/ QUOTE ]

I have perfect scores on the aforementioned tests, a couple of top 200s on the putnam, a 98th percentile score on the GRE math subject test, am a math graduate student, and think that you are very, very wrong with this assertion.

I think that most people who could do this could probably get a degree in math from an average school and possibly a masters from a mediocre school, but a Ph.D is too ambitious.

[/ QUOTE ]

Well firstly I said that they could get their perfect score at age 16. And I stipulated a gun to their head.

Secondly even if you are right, you have merely raise the bar, rather than disputed my point. Maybe it is 15% vs 1% rather than 80% vs 20%.

[furyshade]

is it just me or does every thread in this forum get derailed within 20 posts of the OP?

[hitch1978]

[ QUOTE ]
is it just me or does every thread in this forum get derailed within 20 posts of the OP?

[/ QUOTE ]

No, 7.

[blah_blah]

[ QUOTE ]
is it just me or does every thread in this forum get derailed within 20 posts of the OP?

[/ QUOTE ]

is it just me or is every post you make in math related threads horrible (stellar example: your first post in the thread)

[blah_blah]

[ QUOTE ]
Well firstly I said that they could get their perfect score at age 16. And I stipulated a gun to their head.


[/ QUOTE ]

maybe I am reading your post wrong. If you mean the GRE subject math test, then getting it perfect at 16 is virtually impossible and such a person would probably be able to get a Ph.D easily, even by 20 if you 'held a gun to their head'.

conversely the GRE quantitative section is just as easy as the SAT math portion is, and getting it perfect at 16 is not really all that rare. there are tens of thousands of people who could get both perfect at the age of 16 if they wanted to prepare enough, but the vast majority of them are not Ph.D material.

[pzhon]

I think you are overestimating how difficult it is to get a doctorate in mathematics. It is tough at a good school. It's not nearly as hard if you go to a bad school and are willing to spend a very long time.

I was shocked when I found out the median time to complete a doctorate in mathematics in America. At that time, it was 9 years, but now it is 8, according to the NSF. No one I knew had ever taken that long, and many good schools do not provide support for that long, but but half of the mathematics students who finish take 8 years or longer. (I have since met a professor of mathematics who took 19 years to finish. He really liked graduate school.)

[blah_blah]

the problem with this line of reasoning is that it is more about 'how bad are the worst degree-mills in america' than about the real question that DS is asking, which is 'can a person who is proficient at basic, plug and chug math get an advanced credential in mathematics if they are motivated enough'. almost every decent school I have heard of will require some sort of justification to continue Ph.D studies beyond 7 years, at most. At my university the time limit to finish a Ph.D is 6 years, and you have to advance to candidacy before 36 months or you will be forced to withdraw. I believe that you can extend the second deadline due to extenuating circumstances but not the first.

of course your point is well taken, but I really have no idea as to exactly how poor the quality of scholarship at the worst Ph.D granting institutions in the US is.

[pzhon]

[ QUOTE ]
the real question that DS is asking, which is 'can a person who is proficient at basic, plug and chug math get an advanced credential in mathematics if they are motivated enough'. ... I really have no idea as to exactly how poor the quality of scholarship at the worst Ph.D granting institutions in the US is.

[/ QUOTE ]
I'm not talking about the worst. I'm talking about the quiet majority. It could be that Sklansky means something other than just getting a Ph.D., of course.

[Max Raker]

[ QUOTE ]

(Even though most people like to pooh pooh the SAT and GRE math aptitude tests, I would guess that 80% of those who could get a perfect score on those tests at age 16 could go on to get a Phd in math if a gun was held to their heads while only perhaps 20% of those scoring 700 at age 18 could do that Phd. 700 still being in the top 3% or so. If I'm right it proves that there is an very high correlation between ability at simple math and higher math.)

[/ QUOTE ]

If you mean the GRE math that everyone takes, you are wrong. Getting a 700 means you missed maybe 2 problems (due to how the test is graded and given that many people get every question right) Are you really so results based that you think 2 years and 2 problems changes things by that much?

If you are talking about the math subject test you are wrong again because all who get 800 at 16 and 700 at 18 are 100% able to get a Phd in a gun to head case.

[chezlaw]

[ QUOTE ]
the problem with this line of reasoning is that it is more about 'how bad are the worst degree-mills in america' than about the real question that DS is asking, which is 'can a person who is proficient at basic, plug and chug math get an advanced credential in mathematics if they are motivated enough'.

[/ QUOTE ]
Even though there may be some moderately intelligent people who wont be able to get PHDs but can ace plug and chug maths with a fair amount of study, most of the people who ace these exams are highly intelligent and good candidates for PHDs.

I think DS is underestating the number who can get PHDs from a good university because most of those who ace these plug and chug exams do so with very little effort and are highly intelligent.

chez

[gumpzilla]

[ QUOTE ]
I think you are overestimating how difficult it is to get a doctorate in mathematics. It is tough at a good school. It's not nearly as hard if you go to a bad school and are willing to spend a very long time.

[/ QUOTE ]

Putting it slightly different, I'd say getting a Ph.D in anything is more a question of personality than intelligence.

[David Sklansky]

[ QUOTE ]
[ QUOTE ]

(Even though most people like to pooh pooh the SAT and GRE math aptitude tests, I would guess that 80% of those who could get a perfect score on those tests at age 16 could go on to get a Phd in math if a gun was held to their heads while only perhaps 20% of those scoring 700 at age 18 could do that Phd. 700 still being in the top 3% or so. If I'm right it proves that there is an very high correlation between ability at simple math and higher math.)

[/ QUOTE ]

If you mean the GRE math that everyone takes, you are wrong. Getting a 700 means you missed maybe 2 problems (due to how the test is graded and given that many people get every question right) Are you really so results based that you think 2 years and 2 problems changes things by that much?



[/ QUOTE ]

In my day the test contained 90 questions, 800 meant zero or one wrong (since changed to 5 or six wrong, I think) and 700 meant about FIFTEEN wrong.

[blah_blah]

I believe that an 800 on the quantitative section now is about 89-91st percentile. I am not sure how many questions this corresponds to missing.

[borisp]

[ QUOTE ]
[ QUOTE ]

Even though most people like to pooh pooh the SAT and GRE math aptitude tests, I would guess that 80% of those who could get a perfect score on those tests at age 16 could go on to get a Phd in math if a gun was held to their heads

[/ QUOTE ]

I have perfect scores on the aforementioned tests, a couple of top 200s on the putnam, a 98th percentile score on the GRE math subject test, am a math graduate student, and think that you are very, very wrong with this assertion.

I think that most people who could do this could probably get a degree in math from an average school and possibly a masters from a mediocre school, but a Ph.D is too ambitious.

[/ QUOTE ]
I second this post.

And regarding DS's opinions, etc, here is a blog post from someone who is far more qualified than any of us to discuss this particular subject.

Terrence Tao blog post (http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/)

Granted, it does not address DS's claim directly. But I think it clearly illustrates that he overestimates the impact of "innate intelligence" on the potential success of a mathematician.

As I see it, the most important part of DS's point is the "gun to the head" assumption. As this assumption is wholly unrealistic, I believe DS has simply illustrated that the most important aspect of success in any research endeavor is the willingness to persevere.

It occurs to me that DS lacks this quality with respect to advanced mathematics, etc., and (as is the case with most issues of personal cognitive dissonance) tends to downplay its role in success.

[blah_blah]

I love terence tao's blog to death, but I don't necessarily think that people who win IMO gold medals at the age of thirteen are necessarily the most accurate judges of the value of innate intelligence. for that matter, no one person can be, but it's clear that terence tao's mind works on a completely different level from the rest of ours.

in fact, terence tao provides an affirmative answer to 'can a sufficiently smart person get a Ph.D from the best math school in the world at the age of 20?'

[madnak]

Doesn't the same apply to you? Is Terence Tao really that much smarter than you? I don't mean to be coy, but I would imagine that 99% of math PhDs are incapable of getting top 200 Putnam scores.

[borisp]

[ QUOTE ]
in fact, terence tao provides an affirmative answer to 'can a sufficiently smart person get a Ph.D from the best math school in the world at the age of 20?'

[/ QUOTE ]
Agreed, in general. But I think this particular article highlights exactly what DS is ignoring in his "anyone who is smart could be really smart if they wanted to" posts.

[borisp]

[ QUOTE ]
I would imagine that 99% of math PhDs are incapable of getting top 200 Putnam scores.

[/ QUOTE ]
Is this a joke? Try to imagine what percentages mean...

[bigpooch]

Intelligence or high scores on psychometric/aptitude tests
obviously have some relevance, but one of the most
important aspects MUST BE persistence or perseverance.

There are so many other important factors that are relevant
that this thread hasn't even mentioned!

I personally know of three people who were smart and were in
a Ph.D. mathematics related program at an Ivy League school:
call them S (my brother), M (a friend from university) and
B (a friend from work). All were smart, scored well in
standardized testing, but only one succeeded at getting a
Ph.D. (but obviously none of them won the Putnam or IMO,
even though they scored "well" in them) in a math related
field.

S was in Pure Mathematics specializing in algebraic geometry
at Columbia but quit because he couldn't see the financial
justification of spending more years and money without any
clear career in sight. He is now working for a small
(between 100 and 1000 employees) IT firm with a very
comfortable salary. S was married and had other priorities.

M was in Applied Mathematics but then transferred to
Mathematical Physics at Princeton. The one characteristic
that separated him from all of us was his persistence: he
could be easily up at midnight studying or checking some
calculations. It would be no exaggeration to say that he
could spend eighty hours a week pursuing his interests! He
eventually did get his Ph.D. (of course, it's an easier
field than pure mathematics, right?) in less than six years.
On the other hand, he isn't making a ton of money now and as
far as I know, he is a bachelor.

B was the smartest of the three. He was in Applied Math at
Brown, but his funding had expired. He eventually went to
work for an investment bank ("boutique") and because of the
ridiculous hours (more than 80 hours a week), he then went
to work at an actuarial department in a large insurance
company. For him, being married was also a big factor. He
was also one of the few people who immediately saw how
beautiful Furstenberg's proof of the infinitude of primes
was, but he could also be extremely practical. He was the
least persistent of the three, but he is now doing very well
financially.

[David Sklansky]

"Granted, it does not address DS's claim directly. But I think it clearly illustrates that he overestimates the impact of "innate intelligence" on the potential success of a mathematician."

Suppose the following experiment was conducted where failure was the end of the earth. You and I randomly choose twenty people. You choose from math Phds who got their degreem from Arizona State, Forida State, and Tulane. I randomly choose from Physics, Chemistry, Computer Science, and Economic Phds from Harvard, Cal Tech, and MIT. We all die unless one of them, within the next ten years solves a major, yet unsolved, math problem.

You don't think that my guys inate intelligence makes them 100 times more likely to pull that off than one of your guys?

[bigpooch]

If you added a hundred random PhDs in math to the group of
people that would normally be eligible to write the Putnam,
it would be TOTALLY unfair! I don't think more than two
PhDs would score ZERO, a common score on the Putnam and that
among the top 200 scores, AT LEAST 40 (likely many more!)
would be from the PhD group.
[ Of course, this is just my "silly guess"! /images/graemlins/smile.gif ]

[blah_blah]

[ QUOTE ]
Doesn't the same apply to you? Is Terence Tao really that much smarter than you? I don't mean to be coy, but I would imagine that 99% of math PhDs are incapable of getting top 200 Putnam scores.

[/ QUOTE ]

This is a difficult thing to quantify, so anedcotes and conjecture will have to suffice. I have significantly better problem solving skills than your average graduate student in math, but the difference between me and a Putnam fellow or an IMO gold medalist (I know a couple of each) is huge. If I devoted the next two years of my life to preparing for the Putnam, I suspect that I could break into the bottom of the top 50 or so (even this is optimistic). I would still have zero chance of being a Putnam fellow (yet freshman Putnam fellows are not rare!). The difference between your average Putnam fellow/IMO gold medalist and Terry Tao is probably bigger. He does groundbreaking work in so many different areas of mathematics that it is mind boggling.

[borisp]

[ QUOTE ]
"Granted, it does not address DS's claim directly. But I think it clearly illustrates that he overestimates the impact of "innate intelligence" on the potential success of a mathematician."

Suppose the following experiment was conducted where failure was the end of the earth. You and I randomly choose twenty people. You choose from math Phds who got their degreem from Arizona State, Forida State, and Tulane. I randomly choose from Physics, Chemistry, Computer Science, and Economic Phds from Harvard, Cal Tech, and MIT. We all die unless one of them, within the next ten years solves a major, yet unsolved, math problem.

You don't think that my guys inate intelligence makes them 100 times more likely to pull that off than one of your guys?

[/ QUOTE ]
This is correct, in theory only. This situation would never arise in reality. This is my point.

DS, you are eager to post regarding cognitive dissonance, but you are reluctant to acknowledge when you are engaging in the very practice that you admonish...discuss??

[blah_blah]

[ QUOTE ]
among the top 200 scores, AT LEAST 40 (likely many more!)
would be from the PhD group.
[ Of course, this is just my "silly guess"! /images/graemlins/smile.gif ]

[/ QUOTE ]

I don't think this is correct, the number is less than 20 I think, and may be closer to 10. Most Ph.D's do not have experience with putnam type problems and the average intelligence of your 100 random Ph.D's is probably a fair bit lower than the average intelligence of the top 200 Putnam contestants.

The hypotheticals in this thread are starting to give me a headache /images/graemlins/crazy.gif

[madnak]

[ QUOTE ]
This is a difficult thing to quantify, so anedcotes and conjecture will have to suffice. I have significantly better problem solving skills than your average graduate student in math, but the difference between me and a Putnam fellow or an IMO gold medalist (I know a couple of each) is huge. If I devoted the next two years of my life to preparing for the Putnam, I suspect that I could break into the bottom of the top 50 or so (even this is optimistic). I would still have zero chance of being a Putnam fellow (yet freshman Putnam fellows are not rare!). The difference between your average Putnam fellow/IMO gold medalist and Terry Tao is probably bigger. He does groundbreaking work in so many different areas of mathematics that it is mind boggling.

[/ QUOTE ]

I guess I don't know enough about these things to have a strong concept of what you mean. I was under the impression that only the top 3 mathematicians at a given school are even eligible for the Putnam exam, and that most of them still fail to solve any of the problems correctly. To consistently place in highest 200 in such a competition seems very significant. The gap between Tao and you might be large, but isn't the gap between you and a typical grad student even larger?

More pertinently, are you close enough to the average grad student that I should trust your opinion on the subject more than Tao's?

[Enrique]

[ QUOTE ]
[ QUOTE ]
This is a difficult thing to quantify, so anedcotes and conjecture will have to suffice. I have significantly better problem solving skills than your average graduate student in math, but the difference between me and a Putnam fellow or an IMO gold medalist (I know a couple of each) is huge. If I devoted the next two years of my life to preparing for the Putnam, I suspect that I could break into the bottom of the top 50 or so (even this is optimistic). I would still have zero chance of being a Putnam fellow (yet freshman Putnam fellows are not rare!). The difference between your average Putnam fellow/IMO gold medalist and Terry Tao is probably bigger. He does groundbreaking work in so many different areas of mathematics that it is mind boggling.

[/ QUOTE ]

I guess I don't know enough about these things to have a strong concept of what you mean. I was under the impression that only the top 3 mathematicians at a given school are even eligible for the Putnam exam, and that most of them still fail to solve any of the problems correctly. To consistently place in highest 200 in such a competition seems very significant. The gap between Tao and you might be large, but isn't the gap between you and a typical grad student even larger?

More pertinently, are you close enough to the average grad student that I should trust your opinion on the subject more than Tao's?

[/ QUOTE ]

I agree with blah blah. I am also a top 200 Putnam guy and I was successful in the math olympiad. If I do the Putnam now (I am a mathematics grad student) I would still crack the top 200, but probably not the top 50. The guy that said that if 100 PhDs are allowed to take the Putnam, they would crack the top 200 is wrong in my opinion (at least if they are picked randomly). There is some natural talent.

Also, Terence Tao is a prodigy. He was a gold medalist at the IMO at age 13. His results in mathematics are outstanding. He has results in all sorts of areas.

The Putnam can be taken by anyone in a USA/Canada school. A university can only have 3 on its team, but more people can participate. You can check how MIT has 9 people in the top 20 and 3 in the top 5, yet they didn't win the competition, because the 3 they selected for their team, weren't the best 3. I don't remember if Harvard or Princeton won this year. It always amazes me how MIT has so many students in the top places in the Putnam.

Anyway, I agree that perseverance and hard work are more important than innate ability, but the ability is also important.