TITLE: How do you select an interesting and reasonable problem for a student?
QUESTION [79 upvotes]: I am interested in how to select interesting yet reasonable problems for students to work on, either at Honours (that is, a research-based single year immediately after a degree) or PhD. 
By this I mean a problem that is unsolved but for which there is a good chance that a student can solve it either completely or partially and come out with a thesis either way.
There are a number of possible strategies that I see some of my colleagues use, but which are sadly not available to me:
(1) Be sufficiently brilliant that you already know roughly how something unsolved should be solved and guide the student accordingly, modifying the strategy on the fly.
(2) Have a major project, say classifying a big class of structures, that is amenable to attack with a big general theorem with many well-defined sub-cases that can be assigned individually to students. 
Personally I often work on problems that lead nowhere - I don't solve it, it's too hard, I only rediscover known examples etc - but provided at least some of the problems work out - it doesn't matter.
But for students, its more of a "one-shot" affair - they can't afford to work for a year or, worse, three years, and get nothing.
Are there any general principles that will help in the selection of problems? Or is just a situation where you've either "got the knack" or you haven't?

REPLY [2 votes]: I have advised just one honors student, and in the process of advising another one. However, in my field, (combinatorics) there are some problems that are great for students:

Anything with computers and computation. Students are great at finding patterns. This approach paired with OEIS is a nice combination.
My current student is looking at $q$-analogues of certain counts. Pick your favorite family of combinatorial objects, and see if it has a natural $q$-analogue. This is perfect for a small thesis.
If you feel like the student can handle it and you are lucky with your $q$-analogue, it might have a cyclic sieving phenomena, which one can explore.

Even computer experiments can be reported on, and enumerative problems can usually be "tuned" where special cases can be attacked first.
In my concrete example, we started looking at border-strip decompositions, and counting these. My student quickly came up with a nice family which admits a different interpretation, and fulfills some $q$-analogue properties.