A question about formulation of problems in mathematics

[flamengo]

I have a simple question. Is a mathematics professor able to formulate more difficult problems than the problems that are in the books ? If so, how difficult is it to create such problems ? Is this a difficult task ? Or is this something easy that comes naturally for a professor ? Could someone give me a detailed explanation ?

 

[FactChecker]

Problems in textbooks should be examples that can be solved with the methods shown in the textbook. Any professor should be able to think up difficult examples like that if he wanted to.
Problems in puzzle books may be a lot harder, but should be solvable. The method of solution may not be easy to discover. Puzzles take a special skill that even a very good professor might not have.
Problems in other math books may be famous unsolved problems. It would be extremely difficult for any professor the make up a truly new unsolvable problem. They would become famous.

 

[flamengo]

How do professors develop this ability of formulating problems that are more challenging ?

 

[Demystifier]

Enybody can create a problem that he/she cannot solve or must work hard to solve it. When this is done by professor, then it is a challenging problem almost by definition.

 

[Demystifier]

Could someone give me a detailed explanation ?

Just try to make up a problem that you cannot solve. The things you do during this try, exactly the same is done by the professor (except at a higher level).

 

[S.G. Janssens]

I think that, given a certain field in mathematics, it sometimes requires a deeper understanding of said field to formulate problems that are both challenging and interesting (this is not the same thing, i.m.o.) than to solve them.

Probably this is less valid at the true research level, but I think it is certainly true at the (elementary, intermediate as well as advanced) textbook level.

 

[FactChecker]

Take some regular problems and combine them to make one that takes two or three steps to solve. It's easy.