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Difficulty in problem solving

  1. Dec 2, 2006 #1
    Hi, I am sorry about the length of this but I would appreciate if some you took the time to read and reply.

    I guess I am posting this because I require a little direction at the moment. I am entering into my last year of a math degree in Australia. I have studied quite a bit of math in my degree so far, mostly is has been analysis i.e. calculus, ordinary/partial differential equations, mathematical analysis, complex analysis etc… along with a little algebra/discrete mathematics and statistics. My GPA is 7 which here in the Australian university system is the maximum achievable. I find studying the math at university interesting and easy.

    This all lead to ego driven thoughts that just maybe I could become an eternal figure in mathematics. That I just might be able to solve some exceptionally hard problem and create a place in history for myself. This gave me purpose in life, and not being a religion person it was very comforting to have this purpose.

    Then it all came crashing down. No doubt you have heard of the International Mathematics Olympiad. There is a book called the IMO Compendium and this book contains problems from the previous Olympiads with solutions. High school kids (very talented high school kids) participate in these Olympiads. So I thought these problems would be a fun challenge. Sadly though, the experience turned out to be quite depressing as I found relatively few of them easy, some of them difficult and most of them I just game up on and looked at the solution.

    Here is my problem if I find it difficult to solve these problems, problems that thousands of people in the world can solve (including high school kids) what hope do I have off solving a problem nobody else has solved and receiving my slice of history? I have lost hope and purpose.

    Strangely this experience brings another question to my mind. Why do I find studying mathematics at university so easy but these international mathematical Olympiad problems hard? Is it just lack of exposure to problems of this kind or is it a lack of ability?

    Thanks in advance if you bothered reading this all.
     
  2. jcsd
  3. Dec 2, 2006 #2
    Last edited: Dec 2, 2006
  4. Dec 2, 2006 #3
    I am not 100% sure but I remember hearing somewhere that at some math contest, most students get no questions right. I don't know if that applies to this specific contest as I haven't researched it or anything.

    I used to be not so great at problem solving, mostly because I would just look in the back of the book waaay too fast. What a waste of time that was. After I started to really attempt the problems before looking in the back, I started getting better. I am still not amazing but I find I am better.

    I also found that chess helped my problem solving, it might just be coincidence but I am pretty sure of it.
     
  5. Dec 2, 2006 #4
    It looks like the math contest that I was thinking of is not the IMO.

    I am interested in any suggestions anyone has for improving at problem solving though, I think it is probably just repetively doing problems/puzzles.
     
    Last edited: Dec 2, 2006
  6. Dec 2, 2006 #5
    The thing is I find questions/problems on exams and in the textbooks for the courses I do at university simple yet find these IMO problems hard.

    I wonder if I had a course in "IMO problems" where I had classes and lectures, if that would change the result. Or do the IMO problems just require a higher level of reasoning/abstraction that I may not have the ability for?
     
  7. Dec 2, 2006 #6

    verty

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    Hmm, you have an excellent mark at Uni but it "all came crashing down" because olympiads are tough?

    I wouldn't exactly call it a 'crash'. People are different and some are better at some things than others. For instance, some people have really good memories and can do well in academic environments because they can remember what they need to do, but in an olympiad, I think it is more inventive.

    If I think of a computer algorithm to play chess, one chess algorithm might play very well against one opponent but not well against another. I think it just means the first algorithm is very selective.

    Perhaps your mathematical knowledge is very selective. I wouldn't worry too much about it.
     
  8. Dec 2, 2006 #7
    I have been thinking along similar lines. Maybe I have good technical ability i.e. I can remember techniques to solve learned problems so in an exam when it says solve this PDE I can do it. But I lack a certain creative insight or ingenuity to solve challenging problems.

    This is really the only explanation because the mathematics required to solve the IMO problems isn't complicated but the ingenuity and creative insight required is high.

    This brings me back to my dilemma how can I be a mathematician if I can not solve original problems? I should just be an engineer or something like that but for some reason the only thing I want to be is a mathematician. :grumpy:

    Can “creative insight” be learned or is it something you are born with?
     
  9. Dec 2, 2006 #8
    don't worry about it, the people I met who could solve olympiad problems were no where near the brightest people i ever met. iMO olympiad problems relate more to mathmatical "tricks" more than anything else and spotting those tricks.

    in real math the problems relate to deep intuitive understanding of what is going on. (and with your GPA I'd imagine that you've encountered these kinds of problems before and were able to solve them)
     
  10. Dec 2, 2006 #9

    morphism

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    You do realise that people who do well at olympiad actually train - that is, they practice hundreds and hundreds of olympiad-style problems. So don't lose hope yet!
     
  11. Dec 2, 2006 #10

    verty

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    I think creative insight is something to be learned.

    If you think of creative writing, usually one is given a phrase or topic and one should write something about it. The thing you write should be something that would have been written, given that information. So I think the first task is to take that given topic or pattern and determine where it might fit, then having found where it fits, choose one of the alternatives that best fits the medium; the one that would make the best article/story in the context of your writing assignment.

    Perhaps there are two stories you envision writing (for a given topic) but you can only imagine writing 300 words about one of them and your assignment requires 600 words, you would then choose the longer of the alternatives or try to write a different story.

    So we should say that creativity works in two phases: one first enumerates hypotheses, then iterates over them and chooses the most applicable. Given an olympiad problem, the first step is probably to enumerate all the types of problem it might be or resembles, and then to evaluate each of those hypotheses in turn to determine how to solve it.

    So if your knowledge is very selective, you won't be able to enumerate many hypotheses, and you will inevitably struggle with problems requiring lateral thinking.
     
  12. Dec 2, 2006 #11
    IMO problems are almost impossibly hard.

    Remember that you have more than one hour per problem as well.

    Start out with some simpler problems. I dont know how it works in Australia, but I know that in the US the ladder goes something like this
    AMC-12
    AIME
    USAMO
    IMO

    From experience I can tell you that AIME problems are very difficult, and one of my closer friends qualified for the USAMO (one of the three in Arizona....crazy), and he said those problems are INSANE. I was looking at some of the IMO problems and went blank.

    I'm sure you would be able to do most AMC-12 problems, and a lot of the AIME problems... You can find sample problems online by searching AIME math, or something like that.
     
    Last edited: Dec 2, 2006
  13. Dec 2, 2006 #12
    I just took the Putnam today and most likely got a 0 on it :rofl: so don't feel too bad :smile:
     
  14. Dec 2, 2006 #13

    verty

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    Mathy, this might interest you. From Wikipedia, about George Boole:

     
  15. Dec 2, 2006 #14
    After looking at the solution of the IMO problems I find that a lot of the time I was on the right track and if I had known to use the "trick" that they had used then I would have solved the problem.

    This brings me to another question. Wouldn't solving original difficult math problems that others have struggled with require the use of “tricks”? If we assume that tricks are not needed then the solution is straight forward and somebody would have already solved it. Hence, we must conclude that “tricks” are required in all challenging original problems that have resisted solution.

    So if you can not see the “tricks” behind an olympiad problem what are your chances of seeing a “trick” behind a problem that no one has ever solved before?
     
  16. Dec 2, 2006 #15
    meh, most of math revolves around having the original insight into looking at a problem a certai way.

    for instance if they had a prolem like... create a formula for calculating the tangent to a curve.

    ^before newton created the difference quotient.

    I doubt anybody would be able to solve that problem.

    See real mathematics works around coming up with good questions and finding that unique view point.

    If you look at vector calculus you can see that 90% of that stuff is obvious, its just that nobody thought to put it together.
     
  17. Dec 2, 2006 #16
    A lot of things in mathematics may appear to be intuitively obvious but to prove rigorously they require “tricks”. Newton’s calculus worked but it took hundreds of years for mathematicians to prove and figure out why it worked.
     
  18. Dec 2, 2006 #17
    Yeah, Problem solving is indeed a difficult task. And similarly, when I see little high school kids solving IMO problems that are impossible to me, I was intimidated. But hey, if you can't beat it, learn from it. I started burrowing books from the library and try to learn those "tricks". Gotta tell ya, those are real ingenious and good stuffs.

    Who cares if you are 20 years old or 5 years old. To me, knowledge shouldn't be limited to anyone (although I do feel stupid for not "out-smarting" younger people...but what are you gonna do? start learning why and how they "out-smart" you!) So, why not begin learning these tricks now. New intuitions are always helpful in mathematics.

    BTW, I took the Putnam too... Yeah, it was impossible. I think I got a one though.
     
    Last edited: Dec 2, 2006
  19. Dec 2, 2006 #18
    So your conclusion is that doing well in the IMO just comes down to learning and applying algebraic tricks? I don't know why but I found that a little disconcerting.
     
  20. Dec 2, 2006 #19
    mathy 21 the difference quotient works perfectly well by itself, its a logical extension of the standard definition of algebraic slope.

    now i your trying to derive that its not so much of a trick as it is a matter of thinking about what the change in y really means and what the cahnge in x means.
     
  21. Dec 3, 2006 #20
    I don't mean that doing IMO is merely applying tricks. I mean that the IMO assumes certain knowledge of certain topics. Things like well known theorems, solution to well known problems (Pell's equation, Pythagoras triples). (not that I've been to any one of them)

    If you don't even know Cauchy and AM-GM, how can you be expected to have any insight at all on an Olympiad leveled inequality problem? (unless you are the next gen. Euler of course) After all, competitions winners get trained. They read books, work on problems on a daily basis. Intelligence and hard work are both necessary components of problem solving.
     
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