Hi: I am taking a problem solving class in which we are presented competition math, this class is designed for people writing in the putnam I believe but I am not writing, I just think is pretty fun to solve problems. Anyway, I'm usually only able to solve maybe half of them, then 3/4 takes me all day to think about, and the rest is just impossible to me. I have notice that while these problems require very elegant proof or tactic, sometimes require ideas that I have never thought of( not really new ideas that I haven't learned as in the textbook, but very simple things i never thought of using, but as soon as I was presented to the solution I can agree with it/understand it). that being said, how in the world do people know how to solve these problems so quickly? sometimes to me the solution to these problems are so inconceivable that you have to literally have tried everything possible to find the solution, even for the ones I solved. my strategy is usually just try random things that are at least a bit relevant to the problem , then proceed to assume it will work and prove it does, if i fail, I will move on, and sometimes come back. For those of you good problem solver out there(problems, not exercise), how do you do it? there must be a logical tactic/strategy, or a way of thinking that people use in order to solve these problems. I really don't believe in "you just can't do it"
Hey hihiip201. There is no one answer to your question. It may come down to experience (understanding past problem answers and their relation to the new problem), specific "tricks" that people learn, momentary inspiration, lucky guesses, or particular kinds of insight. One way to think about it is that after a bit of practice, you start to see what to look for and this gets remembered in some form or another.
these reasons make sense, becuz I just think there must be a logical reason as to why some people can see the solution while others don't.
I guess one suggestion would be to get really good at mental math. In elementary, we learn the times table and sadly most average humans can only multiply up to 12 x 12 in their heads. As soon as you throw 13 x 13 at them, they lose their composure. Don't try random things but instead try to find interesting ways to manipulate numbers and handle some computations in your head to save time. That'd be my suggestion. I used to practice doing complex and abstract math without touching a pen or paper. You should try it sometime. It can be fun.
It made solving derivatives and integrals a lot faster, since some of the calculations could be performed in my head. It also made me view mathematics in a different way. When doing mental math I create fragments that are easier to work with. So when I'm approached with a rather daunting problem, I tend to break it up into several smaller problems that are easier to work with. It'd be difficult to give a concrete example, but it's like playing an instrument. You develop a muscle memory, when you first start playing an instrument you have to look at your hands and make sure that your hands are where they need to be. After you get better, you can eventually play without looking. Practicing mental math gave me that sort of muscle memory for math. As stated in my post above, the average non-scientific or math based human will freeze when you ask them to solve 13 x 13 in their heads. 12 x 12 gets taught in elementary and we memorize it. 13 x 13 is a very simple example, but a simple example can go a long way. Instead of trying to solve 13 x 13, I would much rather solve (13 x 10) + (13 x 3) anyone can solve 13 x 10, and 13 x 3 is equally easy. A lot of the mistakes that people make in early Calculus are algebra based. The Calculus is easy, but the algebra confuses people. Being able to solve semi challenging algebra in my head, allowed me to get the basic parts out of the way very quickly before proceeding to tackle the harder parts. The time saved, makes my studying sessions all the more efficient and productive.
The question I asked you was "What made you do problems faster".. You started this answer with "It made solving...". What is "it"? Doing calculations in your head?
Yes, doing calculations in my head made solving problems faster. I elaborated since my previous response was unclear.
I definitely see how doing more mental math can be helpful and efficient at problem solving, as far as textbook problems go. then again, I shouldn't use the word " textbook problem" since it can be anything that can be printed out on a textbook, but what I really want to be able to do is to think outside the box more often. as an example, we were asked a this question in class today: let the area S1 be the area bounded by the y axis, the parabola y=x^2, and y = k where k is some constant (0,1). let area S2 be the area enclosed by x = 1 , the parabola y = x^2 and y = k. at which value of t is the expression S1 + S2 at (1) maximum ? (2) minimum ? many people including myself went ahead and did the whole integral , take derivative, set to zero. when one person just recognized that dS1/dt is just the width (or the upper bound of the area), lower bound for dS2/dt and deduced that the maximum is when the upper bound of S1 = to lower bound of S2. (when the parabola cuts the y = t line in half). ^ that's the kind of thing i want to train myself to do more often. I am able to, not always, but often time I am able to use my out of box brain, but usually not until after staring at a problem for a very long time. another example of me being dumb would be : when answering : how many subsets are there to a set consist of n elements? I said : let's partition the subsets by size..... smart guy: 2^n .....
Memory is silly thing to rely on; it has a habit of being absent when you really need it. You should look at concepts and relationships, and how certain conclusions are derived and obtained. Memorizing 13x13 will not help you at all—as you can derive the solution if needed in a matter of seconds.