Classes and competitions also generally have an allowed 'bag of tricks' from which you build your solutions. White-board coding interviews are the same. If you want to do well, you get to know what the 'atomic' operations are in your bag of tricks, how they combine into useful tools, and how to spot when they might be useful in the 'wild.'
Restating/reworking a problem into a form where some tricks are more obviously applicable is also a useful skill. These are general problem solving skills, which are pretty independent of what's actually in the bag of tricks... Draw a picture, try a concrete example, identify the pen-ultimate step, etc.
When I give whiteboard interviews, I have a favorite question with about four reasonable stopping points. My general approach is to work with the candidate until they get stuck, and then see how handle being stuck and getting un-stuck. So, even if you have trouble with the 'easiest' part of the problem, I can still get a good read on the general problem solving skills. (And if you sail through to the hardest level solution without getting stuck, you're probably worth hiring anyway...)