problem solving

[Manifold]

Hey guys!

(I posted the same thread in physicsforums. To grasp a wider audience I decided to post it here too.)

 

They say the best way to develop and train problem solving skills is by creating one's own problems.

I'm still a "passive" problem solver but I'd like to learn solving problems in an "active" way.

 

Could you give me some advice, name some guidelines on how to create new, own problems?

 

Thanks in advance.

[the tree]

Urm, buy a maths text book? A book of riddles perhaps?

[Manifold]

hm well I think it will at most help training those problem solving strategies...I've got enough books to do that , although you can assume that similar methods would be applicable to few of the problems.

 

I can imagine refining an already solved problem, investigating conditions, constraints, looking for special cases and so on. But the basis for this will be still the problem you already solved.

 

I wonder however if it is the only method of invention in mathematics.

What if I go straight on the "body" of mathematics and try to invent a new problem out of the material I'm working on. I can imagine asking various questions about each theorem, lemma, corollary or whatever. So far it has led me to clarification of the whole picture of how things work in a particular situation, but not to new problems in traditional sense.

[mezarashi]

This is like for example me asking: Someone please teach me how to be a genius ~.~ Now I'm not sure if "genius" is innate or learnt...

[Manifold]

I think it's inappropriate to make such an example in this situation.

I read a number of biographies about prominent mathematicians, who invented problems of their own when they were young, in their pre-university years. They had succeeded in a number of olympiads themselves, so they were good problem solvers. But somehow they managed to make a transition from problem-solvers to problem-inventors. You can't put it down to their assumed genius, can you?

I really think there is more to this than their mathematical ingenuity.

[J.C.MacSwell]

Hey guys!

 

They say the best way to develop and train problem solving skills is by creating one's own problems.

 

I'm still a "passive" problem solver but I'd like to learn solving problems in an "active" way.

 

Could you give me some advice' date=' name some guidelines on how to create new, own problems?[/b']

 

Thanks in advance.

 

 

Tell your wife's best friend you slept with her sister!

[JS]

that's not a problem...

 

tell your best friend you slept with her wife...

 

that's a problem

[Manifold]

ha...ha

[ElijahJones]

If you know what a manifold is you must already have some problem solving skills. You can't get to that level in math without knowing something about proofs. Do you know the logical fallacies? Also if you pick a subject in any of the sciences and study everything about it including all the current literature ofmr the journals you eventually see where the cutting edge is. At this point the subject shows you the outstanding problems (open questions). Number Theory is agood place to make problems for yourself. Pick a Diophantine Equation and find all of its solutions. Be careful it might take th rest of your life depending on what equation you choose. Here I will challenge you with one.

 

Create your own problem by picking a and b and as any integers you like and then solving this Diophantine Equation.

 

[MATH]ax+by^3=1[/MATH]

 

Have fun and please know better than to mention to your frined that you slept with her wife.

 

EJ