> Guessing the right answer in school was always acceptable. Just sitting silently always made the teacher upset.

Yes. This was problematic. I remember my first semester in grad school taking a (for me) tough course. On a midterm, I didn't know how to answer a question, and I thought it was disrespectful to pretend to solve the problem (random relevant equations I could think of) and waste the grader's time with it.

The professor called me into his office and said "Look, I can't give you points if you write nothing. Next time, write something and you'll probably get some partial credit."

>Safer when?

Certainly any time where the work output could potentially harm someone if it's incorrect. Arguably also any time where producing incorrect output or wasting a bunch of time could adversely impact your or a colleague's career. All in all, rather often.

My high school math teacher explicitly designed exams where answers were brief if you recognized the shape of the problem, but you could also just start doing the math and arrive at the answer about 1 page later. It was obvious who understood the problem and who was just stabbing in the dark.

My university CS intro course labs always had an extra "and see if you can do it in O(log n) instead" or such bonus question. The bonus part always "impossible" if you didn't get that a-ha moment.

I think you had bad teachers.. (which is unfortunately much too common)

Unless there's something particularly special going on, presumably you could tell them the truth as it is initially (ie you don't know at that point) and not have that jeopardise dedicating yourself to solving their issue as soon as possible, when you'll give them a final answer.

"I'll find that out for you"