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It's always worthwhile to take some time to try to figure it out by yourself. But eventually there's a point where you're unsure how much more time it would take to figure it out by yourself (which could take far longer than you originally think), and when asking for help could be more helpful.
So for analytical or computational research, what are some guidelines for when you should ask for help (from either your adviser, other grad students, or other researchers) when you get stuck?
I find that grad students always wait too long to ask for help, either out of fear of looking "stupid", or out of inexperience, or out of a misplaced sense of stubbornness. If you've spent a few weeks on something and you're thoroughly stuck, then you should absolutely ask someone (advisor, other students, anyone with knowledge). You'd be surprised at how often you were merely barking up the wrong tree, or just didn't know a very important fact or reference, or just had to reformulate the question differently. All of this takes experience, and that's what an advisor is supposed to provide.
My answer is in two parts.
When to ask other grad students or other researchers for help?
The answer is, anytime. There is nothing to be afraid of asking questions to your office mates, researchers across the doorway or even online. However, they may or may not be able to offer much help because they may not know exactly what you're working on.
When to ask your advisor for help?
The answer is, when you're ready to ask meaningful questions. You don't want your advisor to say why didn't you ask for help sooner. Nor you wanted him to accuse you not working hard enough. If you have meaningful questions and you're stuck for a while - how long is that "a while" usually is a few weeks(long enough to be considered long), you should approach him/her and say you're stuck. Your advisor is there to answer your questions. That's what an advisor is. If you already spend a few weeks on a problem, he would be glad to help you.
You gain more understanding of your research when you try to figure out how to ask questions which make sense because you have to organize your thoughts before asking. Then you're closer to the bottom of it.
However, if you really want to figure it out by yourself for some reasons, you can consider giving yourself a few months. In this case, you would at least let your advisor know you're onto something so that he knows you're making progress. Don't forget he is your advisor. He is on your side !
It took me six months to gather the courage to talk to my research adviser when I was stuck on a project that I knew would "never" work. I should not have waited so long. The conversation we had was very helpful, got me back on track, and, more importantly, got me off the project.
If you are in the same place for more than two or three weeks, its time to have a frank discussion with your research adviser. It's unlikely that he or she knows that you are stuck. The best case scenario is that you have a plan to get unstuck. The worst case scenario is that you are no better off. If you already have a committee, then your next step is to go to the other committee members for help.
It really depends on your advisor's personality, your history of asking for help, the timing of the project, etc. The way in which you ask is also relevant. If you are generally a person who works autonomously and feel you can approach your advisor in a professional (non whiny, non needy) way, most advisors will be happy to help. If you're just afraid to dig in and get your hands dirty (which it sounds you are not), your advisor will probably be less interested in leading you through the process. And there is everything in the middle.
It also depends on whether it's your advisor's project and your project is part of a greater whole; if by delaying too much you are going to slow other people, you should definitely lean on the side of asking for help earlier. However, if you are working on your own idea, you should probably work longer before asking for help. You may also find that you are as much (or more) an expert on the topic than your advisor, so the advisor may not know the answer and may be hoping you'll be the one to figure it out.
There are no general rules. But if you can show that you can figure things out on your own, your advisor may see that as a positive thing and give you a better recommendation for it. On the other hand, if she/he sees you as so stubborn about asking help that you work inefficiently that could be a bad mark.
I do analytical/computational work daily, and I talk to my colleagues quite frequently. Just showing the derivations I made so far to someone often helps me spot my own mistakes, and get new ideas to attack the problem. Your colleagues will often have better understanding of certain subjects that you didn't think would matter for your problem. I guess it depends on your personality (I have highly collaborative), but I would say ask within a day from the moment you got stuck. If someone ends up helping you solve the problem, you will learn from that experience anyways.
You should ask whenever you're stuck. And, as mathematician Peter Sarnak said, "Doing research ... most of the time one is stuck."
You should always be talking to people about what you're working on. You shouldn't be asking them to solve the problem for you, but you should be asking if they know of relevant techniques that might work on your problem.
You should certainly ask your advisor for help every time you meet with him/her. Again, don't ask them to solve the problem for you. Instead, ask whether you should pursue strategy X or technique Y. This can save you a lot of time (but they might be wrong, too).
I constantly discuss my work with my colleges all the time, not only may I get some helpful pointers or references, but asking/staring the question helps me in the solving process, similar to Rubber duck debugging.
