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The vast majority of textbook exercises are worded essentially in the format:

This assertion is (true/false). Prove this or find a counterexample.

This, of course, is not how mathematics is done. In the "real mathematical world", the questions you pose have unknown answers, and thus there isn't such an obvious structure to solving the problem. While it takes some creativity and frustration to come up with a proof that you know exists (only a Google away!), it seems vastly harder to try and actually solve a problem where the definitive answer is unknown.

Since I know many researchers on this site work on such problems every day, I must ask: how does one go about getting started on a problem like this? Do you try to write a positive proof and see where it fails? search for counterexamples by intuition alone?

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Some exercises are stated in terms similar to what you "quote" above, but with the explicit task left to the Reader either find a proof or give a counterexample. So authors sometimes do hold their cards close and leave it to the student to puzzle through the alternatives. –  hardmath Aug 16 '14 at 0:53
Keep in mind that in research you don't have to tackle famous open problems. You can try to find new questions, never asked before, that are interesting yet tractable. Finding good questions can be harder than answering them. –  littleO Aug 16 '14 at 1:00
I wasn't only referring to GRH. I was speaking of any given question that you yourself do not know the answer to. –  theage Aug 16 '14 at 1:02
You should read the book How to Solve it by George Polya. It is one particular set of tactics to attack mathematical problems. –  achille hui Aug 16 '14 at 1:02

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I'm by no means someone who does research, but I don't see this huge difference.

If you find the question on a textbook you have the following theorem at your disposal : "There exists a solution that employs only theorems I've already seen and it is reasonably short and relatively easy to find "

This is an important factor because it keeps you determined and prevents you from giving up too fast, and maybe it will boost your confidence, but if you keep focused it won't be too different.

(Of course, George Dantzig's story is an exception ;-) )

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