Funny enough, I almost exclusively do my best problem solving at night. I'll wake up with solutions, or I'll stay awake and get things done. I'd say 90% of my Master's work was done between midnight and 3am.
But indeed, regardless of time of day, if I just wake up, or am woken up, I'm basically a big, dangerous toddler when it comes to problem solving.
Context and nuance is important, of course. We're all so different.
That often happened for me in grad school as well. Generally the questions I had trouble with on a take home exam would yield to late night inspiration. And if they didn't yield by a semi-reasonable time, I would go to bed and many times, as I was drifting off, an insight would come to me. One memorable time though, that didn't happen and I woke up several times in the middle of the night from stress dreams where I was trying to solve the problem. And when I thought about the dream, nothing I had been doing in it made any logical sense to actually help me with a solution. Fortunately I woke up early and was able to figure it out in the morning. It was not a very restful night of sleep though.
But indeed, regardless of time of day, if I just wake up, or am woken up, I'm basically a big, dangerous toddler when it comes to problem solving.
Context and nuance is important, of course. We're all so different.